analysis of rock performance under three-dimensional

Analysis of Rock Performance under Three-Dimensional

Stress to Predict Instability in Deep Boreholes

Arash Mirahmadizoghi

September 2012

School of Civil, Environmental and Mining Engineering

The University of Adelaide

i

Contents

Abstract ................................................................................................................................. xii

Statement of Originality ....................................................................................................... xv

Acknowledgements ............................................................................................................. xvi

CHAPTER 1 Introduction

1.1. Introduction ..................................................................................................................... 1

1.2. Aims of the Study ............................................................................................................ 4

1.3. Research Method ............................................................................................................. 6

1.4. Organisation of the Thesis ............................................................................................... 8

CHAPTER 2 Literature review

2.1. Introduction ................................................................................................................... 10

2.2. In situ Stresses Prior to the Introduction of the Borehole .............................................. 10

2.3. Stress Analysis around the Borehole ............................................................................. 12

2.4. Strength Analysis of Intact Rock ................................................................................... 16

Coulomb criterion ......................................................................................................... 17

Hoek-Brown criterion ................................................................................................... 18

2.4.1. The influence of intermediate principal stress on rock failure stress ..................... 21

Yield and failure ........................................................................................................... 22

2.4.2. Frictional criteria ..................................................................................................... 25

Drucker-Prager criterion ............................................................................................... 26

Modified Wiebols and Cook criterion .......................................................................... 27

Modified Lade criterion ................................................................................................ 29

2.4.3. Hoek-Brown based criteria ..................................................................................... 31

Pan-Hudson criterion .................................................................................................... 32

ii

Generalised Priest criterion ........................................................................................... 33

Simplified Priest criterion ............................................................................................. 34

Generalised Zhang-Zhu criterion .................................................................................. 35

CHAPTER 3 Stress analysis around a borehole

3.1. Introduction ................................................................................................................... 37

3.2. Stress Analysis around a Vertical Borehole .................................................................. 40

Stresses before drilling the borehole ............................................................................. 40

Stresses after drilling the borehole ................................................................................ 42

Changes in the initial stress state due to the introduction of the borehole .................... 44

Total induced in situ stresses ........................................................................................ 44

3.2.1. Numerical model of a vertical borehole ................................................................. 46

3.3. Stress Analysis around a Deviated Borehole ................................................................. 53

3.3.1. Stresses at the borehole wall due to far-filed in-plane shear, �� and

normal stresses, ��, �� and �� ...................................................................................... 55

3.3.2. Stresses at the borehole wall due to longitudinal shear stresses

(�� ) and (�� ) ............................................................................................. 58

3.4. Numerical Counterpart of the Generalised Kirsch Equations ....................................... 60

3.5. A Modification to the Generalised Kirsch Equations ................................................... 70

CHAPTER 4 Rock strength analysis in three-dimensional stress

4.1. Introduction ................................................................................................................... 76

4.2. Definition of General, Principal and Deviatoric Stress Tensors ................................... 77

4.3. Failure Function in Principal Stress Space .................................................................... 80

4.4. Failure functions in deviatoric stress space ................................................................... 82

4.5. Failure Criteria on Deviatoric and Meridian Planes ...................................................... 87

4.6. Failure Criteria Especially Developed For Rock Material ............................................ 91

iii

4.6.1. The Hoek-Brown criterion ...................................................................................... 92

4.6.2. The Pan-Hudson criterion ....................................................................................... 96

4.6.3. The Zhang-Zhu criterion ......................................................................................... 99

4.6.4. Generalised priest criterion ................................................................................... 102

4.6.5. The Simplified Priest Criterion ............................................................................. 106

4.7. Experimental Evaluations of Rock Behaviour under Three-Dimensional Stress ........ 109

4.7.1. The influence of intermediate principal stress on failure stress ............................ 110

4.7.2. A modification to the simplified Priest criterion .................................................. 116

4.7.3. Comparison of three-dimensional, Hoek-Brown based failure criteria ................ 120

4.8. True-triaxial Experiments at the University of Adelaide ............................................ 124

4.8.1. Experimental setup ............................................................................................... 124

True-triaxial apparatus ................................................................................................ 124

Specimen preparation ................................................................................................. 126

True-triaxial tests ........................................................................................................ 127

Comparison and validation of three-dimensional failure criteria against

true-triaxial data .......................................................................................................... 128

CHAPTER 5 A case study of prediction of borehole instability

5.1. Introduction ................................................................................................................. 136

5.2. Prediction of borehole instability ................................................................................ 137

5.3. Designing the drilling fluid.......................................................................................... 143

CHAPTER 6 Conclusion

6.1. Summary and conclusions ........................................................................................... 147

6.2. Recommendations for future studies ........................................................................... 148

References ............................................................................. Error! Bookmark not defined.

Appendix A The Finite Element Method ......................................................................... 155

iv

The Finite Element Method (FEM) ................................................................................ 155

Mesh quality ............................................................................................................... 157

APPENDIX B Qunatitative comparison between analytical and numerical models ....... 159

APPENDIX C True-triaxial data from the literature ....................................................... 164

APPENDIX D ��-�� plots for the selected rock types from the literature ...................... 174

APPENDIX E Error analysis diagrams ............................................................................ 196

APPENDIX F MATLAB programs for plotting three dimensional failure surfaces in the

principal stress space .......................................................................................................... 201

Hoek-Brown Criterion ................................................................................................ 201

Pan-Hudson Criterion ................................................................................................. 204

Zhang-Zhu Criterion ................................................................................................... 207

Simplified Priest Criterion .......................................................................................... 210

Generalised Priest Criterion ........................................................................................ 213

v

Figures

Figure 1.1 Failure (�) and yield (��) stresses for brittle materials .......................................... 4

Figure 1.2 Demonstration of different phases in the stepwise research method

adopted in this study ................................................................................................................... 6

Figure 2.1 Coordinate system for a deviated borehole [after Fjær et al. (2008)] ................... 12

Figure 2.2 Mean octahedral shear stress, �� vs. mean octahedral normal �� at

yield for Dunham dolomite (after Mogi (2007)) ..................................................................... 24

Figure 2.3 The cross section of (a) the Coulomb, (b) the circumscribed and (c)

the inscribed Drucker-Prager on the deviatoric plane .............................................................. 27

Figure 2.4 Relation between intermediate and major principal stresses at failure

for eight different failure criteria for a rock mass subjected to a minor principal

stress of 15 MPa, with a uniaxial compressive strength of 75 MPa, mi = 19 and

GSI = 90 (Priest, 2010) ............................................................................................................. 33

Figure 3.1 Stresses on an element at a radial distance r from the centre of a

circular hole with radius a, in polar coordinates. ...................................................................... 38

Figure 3.2 The model of the pre-stressed rock block into which the borehole

will be drilled ............................................................................................................................ 41

Figure 3.3 Demonstrating the conditions for applying plane strain assumption

for calculating longitudinal stress components around a borehole…….. ................................. 43

Figure 3.4 Radial distance from the borehole centre and angular position of a

given element ............................................................................................................................ 48

Figure 3.5 Comparison between numerical and analytical model for variation

of induced radial (��) and tangential (��) stresses around the vertical

borehole at � = 0.085 m ............................................................................................................ 50

vi


of induced vertical (��) and in-plane shear (��) stresses around the vertical

borehole at � = 0.085 m ............................................................................................................ 51


of induced stresses along the radial direction r, at � = 0, for the vertical

borehole .................................................................................................................................... 52


of induced in-plane shear stress along the radial direction r, at � = 0, for a

vertical borehole ....................................................................................................................... 53

Figure 3.9 General stress state in the vicinity of an inclined borehole ................................. 54

Figure 3.10 Corresponding stresses for (a) and (b) plain strain problem and (c)

for anti-plane strain problem ..................................................................................................... 55

Figure 3.11 Demonstrating the method adopted for calculating induced stresses

around a borehole due to pure far-field shear stresses, acting on a plane

perpendicular to the borehole axis ............................................................................................ 56

Figure 3.12 Deformations associated with anti-plane strain boundary conditions ............... 60


of induced radial (��) and tangential (��) stresses around the inclined borehole

at � = 0.085 m ........................................................................................................................... 64


of induced vertical (��) and in-plane shear (��) stresses around the borehole at

� = 0.085 m ............................................................................................................................... 65


of induced longitudinal shear stresses �� and ��, around the inclined borehole

at � = 0.085 m ........................................................................................................................... 66


of induced stresses along the radial direction r, at � = 55.166°, for the inclined

borehole .................................................................................................................................... 69

Figure 3.17 A section of a borehole at the depth of 3000 m .................................................. 71

Figure 3.18 Changes in longitudinal shear stresses around the borehole under

the proposed boundary conditions ............................................................................................ 73

vii

Figure 3.19 Changes in longitudinal shear stresses under the proposed

boundary conditions, along the radial direction from the borehole wall .................................. 74

Figure 4.1 Compressive general stresses on a block of rock ................................................ 77

Figure 4.2 Principal stresses on a block of rock ................................................................... 78

Figure 4.3 Failure surface in the principal stress space ........................................................ 81

Figure 4.4 Hydrostatic axis and the stress vector � in the principal stress space ................. 82

Figure 4.5 Deviatoric and �-plane ........................................................................................ 83

Figure 4.6 Cartesian coordinate system on the deviatoric plane .......................................... 84

Figure 4.7 Polar components of point P on the deviatoric Plane ......................................... 86

Figure 4.8 Symmetry properties of a failure criterion on the deviatoric plane ..................... 88

Figure 4.9 Meridional plane (� � � coordinates) [after Ottosen and Ristimna(2005)] ........ 90

Figure 4.10 Intersection of tensile and compressive meridians with the deviatoric

plane .......................................................................................................................................... 91

Figure 4. 11 The cross section of the Hoek-Brown failure surface on the deviatoric

plane .......................................................................................................................................... 93

Figure 4.12 The Hoek-Brown criterion in the principal stress space .................................. 95

Figure 4.13 The cross section of the Hoek-Brown criterion on the deviatoric plane .......... 97

Figure 4.14 The Pan-Hudson criterion in the principal stress space ................................... 98

Figure 4.15 The cross section of the Zhang-Zhu criterion on the deviatoric plane ........... 101

Figure 4.16 The Zhan-Zhu criterion in the Principal stress space ..................................... 102

Figure 4.17 The cross section of the generalised Priest criterion on the

deviatoric plane ....................................................................................................................... 105

Figure 4.18 The generalised priest criterion in the principal stress space ......................... 106

Figure 4.19 The cross section of the simplified Priest criterion on the

deviatoric plane for (a) �� 10 MPa, (b) �� 100 MPa ..................................................... 108

Figure 4.20 The Simplified Priest criterion in the principal stress space, for

(a) �� 10 MPa � � 0.211 and (b) �� 100 MPa, � � 2.99 .......................................... 109

Figure 4.21 Fitting quadratic functions to true-triaxial experimental data in

�� domain (continues) .................................................................................................... 113

viii

Figure 4.22 Non-linear correlation coefficient between the failure stress (��)

and the intermediate principal stress (��) versus the least principal stress (��) ..................... 115

Figure 4.23 Actual values of the weighting factor � versus values of the term

�� 3� .............................................................................................................................. 119

Figure 4.24 Difference between predicted and observed failure stresses .......................... 121

Figure 4.25 True-triaxial apparatus of the University of Adelaide [after

Schwartzkopff et al.(2010)] .................................................................................................... 125

Figure 4.26 Block of Kanmantoo Blue stone and preparation of cubic specimens

[after Dong et al., (2011)] ....................................................................................................... 126

Figure 4.27 (a) The V-shaped failure mode and (b) the M-shaped failure mode

[after Dong et al. (2011)] ........................................................................................................ 128

Figure 4.28 Best fit line to conventional triaxial data for determining the

Hoek-Brown constant parameter m ........................................................................................ 130

Figure 4.29 ��-�� plots, demonstrating that all 3D failure criteria underestimate

the strength of the rock specimen ........................................................................................... 131

Figure 4.30 Intrusion of the HDPE plastic layer into the rock specimen [after

Dong et al (2011)] ................................................................................................................... 132

Figure 4.31 Best fit line to triaxial test data on cubic specimens for determining

the empirical parameter m ....................................................................................................... 133

Figure 4.32 ��-�� plots, demonstrating the comparison of the selected three-

dimensional failure criterion ................................................................................................... 134

Figure 5.1 Principal in situ stresses acting on a rock element at the borehole

wall, with drilling fluid ........................................................................................................... 145

Figure A.1 The numerical error of the observed field variable (in this

case !"#$) can be minimized by increasing the discretisation resolution

stepwise from (a) to (c). .......................................................................................................... 158

ix

Figure C.1 Linear correlation coefficient calculated by the means of Pearson

linear correlation coefficient for the nine sets of true-triaxial data ......................................... 173

Figure D.1 ��%&. �� Plots for KTB Amphibolite for different constant values of �� .......... 174

Figure D.2 ��%&. �� Plots for Westerly Granite for different constant values of �� ............ 176

Figure D.3 ��%&. �� Plots for Dunham Dolomite for different constant values of �� ........ 180

Figure D.4 ��%&. �� Plots for Solnhofen Limestone for different constant values of �� ..... 184

Figure D.5 ��%&. �� Plots for Yamaguchi Marble for different constant values of �� ........ 186

Figure D.6 ��%&. �� Plots for Mizuho Trachyte for different constant values of �� ........... 187

Figure D.7 ��%&. �� Plots for Manazuru Andesite for different constant values of �� ......... 189

Figure D.8 ��%&. �� Plots for Inada Granite for different constant values of �� ................. 191

Figure D.9 ��%&. �� Plots for Orikabe Monzonite for different constant values of �� ......... 194

Figure E.1 Normal distribution of failure prediction accuracy of selected

failure criteria for Orikabe Monzonite .................................................................................... 196


failure criteria for Inada Granite ............................................................................................. 196


failure criteria for Manazuru Andesite .................................................................................... 197


failure criteria for Mizuho Trachyte ....................................................................................... 197


failure criteria for Yamaguchi Marble .................................................................................... 198


failure criteria for Solnhofen Limestone ................................................................................. 198


failure criteria for Dunham Dolomite ..................................................................................... 199


failure criteria for Westerly Granite ........................................................................................ 199


failure criteria for KTB Amphibolite ...................................................................................... 200

x

Tables

Table 3.1 Determining the angular position of the two points of stress

concentration ............................................................................................................................. 68

Table 4.1 Hoek-Brown and Coulomb parameters of the rock types studied ....................... 110

Table 4.2 Comparison of 3D Hoek-Brown based criteria ................................................... 122

Table 4.3 True-triaxial experimental data of Kanmantoo Bluestone, The

University of Adelaide (2011) ................................................................................................ 127

Table 4.4 Uniaxial compressive strength of cylindrical and cubic specimens

of Kanmantoo bluestone. ........................................................................................................ 128

Table 4.5 Conventional triaxial tests for determining the Hoek-Brown constant

parameter m ............................................................................................................................. 129

Table 4.6 Predicted values of failure stress by the means of each selected

failure criteria for m = 16.131 and � = 147 MPa ................................................................... 131

Table 4.7 Triaxial test data on cubic rock specimens for determination the

empirical parameter m ............................................................................................................ 133

Table 4.8 Predicted values of failure stress by the means of each selected

failure criteria for m = 36.6 and � = 190.3 MPa .................................................................... 134

Table 4.9 Error analysis and quantitative comparison of selected 3D failure

criteria ..................................................................................................................................... 135

Table 5.1 Calculation of the failure stress for Granite and Marble ..................................... 143

Table B.1 Error analysis of the finite element model in comparison with the

analytical solution, for calculating the induced stresses around the vertical

borehole (for a quarter-model) ................................................................................................ 159


analytical solution (the generalised Kirsch equations), for calculating the

induced stresses around a deviated borehole (for a quarter-model) ....................................... 160

xi

Table B.3 Error analysis of the finite element analysis based on the proposed

boundary conditions in comparison with the analytical solution (the generalised

Kirsch’s equations), for calculating the induced stresses around a deviated

borehole (for a quarter-model) ................................................................................................ 161


analytical solution (the generalised Kirsch’s equations), for calculating the

induced stresses along the radial distance r from the wall of a deviated borehole

at � = 55.166° ......................................................................................................................... 162

Table B.5 Error analysis of the finite element analysis based on the proposed

boundary conditions in comparison with the analytical solution (the generalised

Kirsch’s equations), for calculating the induced stresses along the radial

distance r from the wall of a deviated borehole at � = 55.166° ............................................. 163

Table C.1 True-triaxial data of Solnhofen Limestone, Mogi (2007) ................................... 164

Table C.2 True-triaxial data on Dunham Dolomite, Mogi (2007)....................................... 165

Table C.3 True-triaxial data on Yamaguchi Marble, Mogi (2007) ..................................... 166

Table C.4 True-triaxial test data on Mizuho Trachyte (Mogi, 2007) .................................. 167

Table C.5 True-triaxial test data on Orikabe Monzonite (Mogi, 2007) ............................... 168

Table C.6 True-triaxial test data on Inada Granite (Mogi, 2007) ........................................ 169

Table C.7 True-triaxial test data on Manazuru Andesite (Mogi, 2007) .............................. 170

Table C.8 True-triaxial test data on KTB Amphibolite (Chang and Haimson,

2000) ....................................................................................................................................... 171

Table C.9 True-triaxial test data on Westerly Granite (Haimson and Chang,

2000) ....................................................................................................................................... 172

xii

Abstract

Underground rock formations are always under some stress, mostly due to overburden

pressure and tectonic stresses. When a borehole is drilled, the rock material surrounding the

hole must carry the load which was initially supported by the excavated rock. Therefore, due

to the introduction of a borehole, the pre-existing stress state in the sub-surface rock mass is

redistributed and a new stress state is induced in the vicinity of the borehole. This new stress

state around the borehole can be determined directly by means of in situ measurements, or can

be estimated by applying numerical methods or closed form solutions.

In this thesis borehole stability analysis is undertaken by means of the linear elasticity theory.

The introduction of a borehole into a block of rock which behaves linearly elastic, leads to

stress concentration near the hole. If the rock material around the borehole is strong enough to

sustain the induced stress concentration, the borehole will remain stable; otherwise rock

failure will occur at the borehole wall. Therefore, a key aspect in stability evaluation of a

borehole is the assessment of rock response to mechanical loading.

For borehole stability evaluation in good quality brittle rock formations, which can be

considered as isotopic, homogeneous and linearly elastic, stresses around the borehole are

usually calculated using a closed form formulation known as the generalised Kirsch equations.

These equations are the three-dimensional version of the original form of the well known

Kirsch equations for calculating stresses around a circular hole in an isotropic, linearly elastic

and homogeneous material. These equations have been widely used in the petroleum and

mining industries over the past few decades. However, the boundary conditions on which

these equations were based have been poorly explained in the literature and therefore merit

further investigation.

In this thesis, in order to eliminate the ambiguity associated with the boundary conditions

assumed for deriving the analytical model for stress analysis around the borehole, finite

element analysis (FEA) was carried out to create a numerical counterpart of the current

analytical solution. It appeared that the assumed boundary conditions for deriving the

analytical model, i.e. the generalised Kirsch equations, are incompatible in the physical sense.

xiii

A new set of boundary conditions in better compliance with the physics of the problem was

introduced in order to modify the analytical model, by reducing the simplifying assumptions

made to facilitate the derivation of the closed form solution.

Another key parameter in borehole stability evaluation is the strength of the rock material at

the borehole wall. The rock strength is usually evaluated using a failure criterion which is a

mathematical formulation that specifies a set of stress components at which failure occurs. A

number of different failure criteria have been introduced in the literature to describe brittle

rock failure among which the Coulomb and the Hoek-Brown criteria have been widely used in

industry; however, they both have limitations. For instance, both the Coulomb and the Hoek-

Brown criteria identify the rock strength only in terms of maximum and minimum principal

stresses and do not account for the influence of the intermediate principal stress on failure. On

the other hand, at the borehole wall where a general stress state (�� ' �� ' ��) is encountered,

a failure criterion which neglects the influence of the intermediate principal stress on failure

seems to be inadequate for rock strength estimation in the borehole proximity.

Although a number of three-dimensional failure criteria have been proposed over the past

decades, none of them has been universally accepted. A major limitation in studying the three-

dimensional rock failure criteria is the lack of adequate true-triaxial experimental data that can

be used for validation of theoretical rock failure models. A number of true-triaxial tests were

carried out at the University of Adelaide and the results, along with nine sets of published

true-triaxial experimental data, were utilised for comparison and validation of five selected

failure criteria. These failure criteria have been developed especially for rock material and

include; the Hoek-Brown, the Pan-Hudson, the Zhang-Zhu, the Generalised Priest and the

Simplified Priest. A new three-dimensional failure criterion was also developed by modifying

the simplified Priest criterion and was identified as a three-dimensional model which best

describes the rock failure in three-dimensional stress state, compared to other selected criteria.

In this thesis, a case example is presented where the borehole instability is predicted by

comparing the induced major principal stress at the borehole wall to the predicted rock failure

stress. The major in situ principal stress around the borehole is calculated by means of the

FEA based on the assumption of a new set of boundary conditions. The rock failure stress

xiv

under the three-dimensional stress state at the borehole wall is calculated by means of the

newly proposed three-dimensional failure criterion.

xv

Statement of Originality

This work contains no material which has been accepted for the award of any other degree or

diploma in any University or other tertiary institution and, to the best of my knowledge and

belief no material previously published or written by any other person, except where due

reference has been made in the text.

I give consent to this copy of my thesis, when deposited in the University Library, being made

available for loan and photocopying, subject to the provisions of the Copyright Act 1968.

I also give permission for the digital version of my thesis to be made available on the web, via

the University’s digital research repository, the Library catalogue, and also through web

search engines, unless permission has been granted by the University to restrict access for a

period of time.

SINGED:............................................ DATE:........................................

xvi

Acknowledgements

The work described in this thesis was carried out in the School of Civil, Environmental and

Mining Engineering at the University of Adelaide during the period 2010 to 2012. The

candidate was supervised for the first one and a half years by Professor Stephen D. Priest and

Dr. Nouné S. Melkoumian and, after Professor Priest retired, by Dr. Nouné S. Melkoumian

and Associate Professor Mark B. Jaksa. The author is indebted to Professor Stephen D. Priest

for providing the opportunity for this research to be carried out and for seeing it to fruition.

Much appreciation is also shown to Dr. Nouné S. Melkoumian and Assoc. Prof. Mark B. Jaksa

for being generous with their advice, assistance and guidance. The author is grateful for the

discussions with Assoc. Prof. Hamid Sheikh and for his technical advice on finite element

analysis presented in Chapter 3 of this thesis.

The author wishes to acknowledge the support of the Deep Exploration Technologies CRC for

partially funding the true-triaxial test and the ABAQUS licenses used in this research. No

amount of thanks would be enough to give to the technical staff in the School of Civil,

Environmental and Mining Engineering at the University of Adelaide, for without their

assistance and encouragement conducting the true-triaxial tests would have been impossible.

In particular the author would like to give special mention to:

• Mr. David Hale, the manager of technical operations, for providing invaluable

comments and advice for true-triaxial experiments.

• Mr. Adam Ryntje for preparation of rock specimens for true-triaxial tests and for his

assistance in operating the true-triaxial apparatus.

• Mr. Ian Cates for supervising the laboratory instrumentation and running the data

acquisition system and for his assistance in collecting the true-triaxial test data.

• Dr. Stephen Carr for his help in running ABAQUS models associated with the finite

element analysis presented in Chapter 3.

I wish to express my appreciation to my mother and my wife for without their financial

and mental supports conducting this research would be impossible.

CHAPTER 1

Introduction

Page

1.1. Introduction 1

1.2. Aims of the study 4

1.3. Research method 6

1.4. Organisation of the thesis 8


1

1.1. Introduction

Borehole stability problems have been encountered for as long as wells have been drilled.

Several new challenges have appeared in recent years, however, making the stability issue

more difficult to handle, and also more important to solve. For example, there has been an

increasing demand from the petroleum industry for more sophisticated well trajectories.

Highly deviated, multilateral and horizontal wells are attractive to the petroleum industry,

since a single production platform supporting a sophisticated well can drain a larger area,

reducing the number of platforms required to produce a given field (Fjær et al., 2008). Stable

drilling is however normally more difficult in deviated than in vertical boreholes. Other

situations where borehole stability problems may be expected to occur are during infill drilling

in depleted reservoirs, when drilling in tectonically active areas, and in deep and geologically

complex surroundings (Zoback, 2007).

A borehole stability problem is an example of what drillers refer to as a “tight hole” or “stuck

pipe” incident. There are many possible reasons for drilling rigs to become stuck, but in the

majority of field cases reported, the fundamental reason is the mechanical collapse of the

borehole wall (Bol et al., 1994, Gazaniol et al., 1994). Moreover, the mechanical collapse of

the borehole wall is often combined with a lack of down-hole cleaning ability. Such stability

problems typically amount to 5%–10% of drilling costs in exploration and production, in

terms of time lost and sometimes loss of equipment. These numbers imply a worldwide cost to

the petroleum industry of hundreds of millions of dollars per year (Fjær et al., 2008).

Underground formations are always under some stress, mostly due to overburden pressure and

tectonic stresses. When a borehole is drilled the rock material surrounding the hole must carry

the load which was initially supported by the excavated rock. Therefore, the pre-existing stress

state in the sub-surface rock mass is redistributed and a new stress state is induced in the

borehole proximity. This new stress state around the borehole can be determined directly by

applying in situ measurements, numerical methods or closed form solutions. The introduction

of a borehole into a block of rock which can be considered as linearly elastic, leads to stress

concentration near the hole. If the rock material around the borehole is strong enough to


2

sustain the induced stress concentration the borehole will remain stable. On the other hand, if

the stresses around the borehole exceed the strength of the surrounding rock, failure will

eventually occur and a fractured zone will develop around the hole. If the fractured zone is too

extensive either the design of the excavation must be modified or the excavation must be

supported appropriately.

A key aspect in the stability evaluation of a borehole, therefore, is the assessment of the rock

response to mechanical loading. Ideally a technical model should account for all factors which

could affect stability, such as well pressure, temperature, time and mud chemistry; however,

such a model is currently unavailable. The focus of the current study will be on the effect of

rock response to mechanical loading on borehole stability

For borehole stability evaluation in good quality brittle rock formations, which can be treated

as isotopic, homogeneous and linearly elastic, a two-step analysis is suggested, which consists

of:

1. Calculating the stresses around the borehole using the linear elastic theory.

2. Assessing the strength of the rock material surrounding the borehole under the induced

stress state due to drilling the borehole.

Calculating stresses: When a borehole is drilled into an ideal rock block (isotropic, linearly

elastic and homogeneous) stresses around the borehole are usually calculated using a closed

form formulation known as the generalised Kirsch equations. These equations are a three-

dimensional version of the original form of the Kirsch equations (1898) for calculating

stresses around a circular hole in an isotropic, linearly elastic and homogeneous material. The

three-dimensional version of Kirsch equations can be found in a report by Fairhurst (1968).

These equations have been widely used in the petroleum and mining industries over the past

few decades. However, the boundary conditions on which these equations were based have

been poorly explained in the literature and therefore merit further investigation.

Assessing rock strength: Another key parameter in borehole stability evaluation is the

strength of the rock material at the borehole wall. The rock strength is usually formulated as a

failure criterion, which is a mathematical formulation that specifies a set of stress components


3

at which failure occurs. A number of different criteria have been introduced in the literature to

describe brittle rock failure, among which the Coulomb, introduced in 1773, and the Hoek-

Brown (1980) criteria have been widely used in industry; however, each has some limitations.

For instance, both the Coulomb and the Hoek-Brown criteria identify the rock strength only in

terms of maximum and minimum principal stresses and do not account for the influence of the

intermediate principal stress on failure. On the other hand, at the borehole wall where a

general stress state (�� ' �� ' ��) is encountered, a failure criterion which neglects the

influence of the intermediate principal stress on failure seems to be inadequate to rock strength

estimation in the borehole proximity.

Nevertheless, currently in the petroleum industry, estimation of rock strength around the

borehole is mostly undertaken by applying either the Coulomb criterion, or the Drucker-Prager

criterion (1952), which is a three-dimensional failure criterion and incorporates the influence

of the intermediate principal stress. However, the Drucker-Prager criterion was initially

developed for soil. The results of true-triaxial rock testing show that this criterion is unable to

accurately predict rock strength under a three-dimensional stress state when compared with the

three-dimensional criteria which have been developed especially for rock material

(Colmenares and Zoback, 2002). It is desirable, therefore, to use a three-dimensional failure

criterion specifically designed for rock material to predict rock strength.

Although a number of three-dimensional rock failure criteria have been proposed over the past

few decades, none has been universally accepted. A major limitation for studying three-

dimensional rock failure criteria is inadequate true-triaxial test data on rock specimens and as

pointed out by Mogi (2007) the few published results of true-triaxial tests have been

interpreted more as interesting curiosities, rather than serious challenges to the accepted Mohr-

type criteria. Therefore, three-dimensional rock failure criteria require further investigation,

incorporating more comprehensive true-triaxial experimentation.

After calculating the induced stresses around the borehole and predicting the failure stress

using an appropriate rock failure criterion, which best describes the rock behaviour -under a

three-dimensional stress state-, the possibility of the failure of the rock at the borehole wall

can be estimated. It is worthwhile emphasising that in this study borehole stability analysis is


4

undertaken by means of the linear elasticity theory. Therefore, strength evaluation of intact

brittle rock material, which can be assumed to be continuum, homogeneous, isotropic and

linearly elastic (CHILE), is of interest in this study. The portion of inelastic behaviour, in

stress-strain plots, before failure, is small in brittle rocks and an abrupt failure occurs shortly

after the elastic limit or the yield stress, �� is reached (Fig. 1.1). Therefore, since yield and

failure for brittle materials are approximately the same, it is theoretically rational to compare

the value of the maximum induced stress at the borehole wall estimated by means of linear

elasticity theory to the failure stress of the brittle rock material. It also merits mentioning that

in the case of ductile materials since the failure stress is significantly greater than the elastic

limit, linear elastic analysis is a conservative model for predicting borehole failure.

Furthermore, failure progression as the accumulation of damage and non-linearly elastic

behaviour of rock is beyond the scope of this study.

1.2. Aims of the Study

Boundary conditions based on which stresses around the borehole were formulated, in the

generalised Kirsch equations, have been poorly explained in the existing literature and need to

be studied in further detail. It is important to remember that some simplifying assumptions

were made about the constitutive behaviour of the rock material to rationalise the application

of the Kirsch equations for rock material. In other words the rock material is assumed to be a

continuum, homogeneous, isotropic and linear elastic material. Even for the linear elastic

� ��

�

(

Failure

Yield

Figure 1.1 Failure ()*) and yield ()+) stresses for brittle materials


5

constitutive behaviour solving a general three-dimensional problem may be impossible.

Therefore, boundary conditions are defined, considering the physics of the problem, in a way

to facilitate the derivation of an analytical model. However, the boundary conditions assumed

in the current analytical model do not seem to be in compliance with the physics of the real

problem and need modifications.

On the other hand, since the rock material at the borehole wall is subjected to a three-

dimensional stress state (�� ' �� ' ��), a two-dimensional failure criterion such as the

Coulomb or the Hoek-Brown is inadequate for estimating rock strength in the borehole

proximity. Therefore, in order to predict the rock failure at the borehole wall, three-

dimensional rock failure criteria need to be given further consideration.

Aims of this thesis are outlined as follows:

- Providing a clear explanation to eliminate the ambiguity about the boundary conditions

assumed for deriving the current analytical model, i.e. the generalised Kirsch

equations.

- Modification of the current analytical model by introducing a new set of boundary

conditions, which reflects the physics of the problem more realistically.

- Theoretically investigating a number of selected three-dimensional rock failure criteria

through conceptual studies on failure theory.

- Development and introduction of a new three-dimensional rock failure criterion.

- Conducting true-triaxial experiments to validate and compare the existing and the

newly proposed rock failure criteria against the true-triaxial experimental data.

- Demonstration of the linkage between techniques of the stress analysis in the borehole

proximity and the strength estimation of rock material adjacent to the borehole wall

through a case study, by predicting instability of a deviated borehole and calculating

the minimum and maximum allowable mud weight to safely drilling the deviated

borehole.


6

1.3. Research Method

In order to evaluate the stability of deep boreholes a stepwise analysis is adopted. This

stepwise analysis is depicted in the flow chart in Fig. 1.2 and the aims of this research in the

frame of this analysis are subsequently discussed.

Step A Linear elastic stress analysis is carried out to estimate the induced stresses around the

borehole by means of numerical and analytical models. Results of the numerical and analytical

models are cross-evaluated to investigate the boundary conditions assumed in formulating the

A B

C

Borehole Stability Evaluation

Stress analysis around the borehole Estimation of the strength of rock

materials surrounding the borehole

Comparison between estimated stresses and predicted rock strength in the vicinity of the borehole

for evaluation of the borehole stability

Devising strategies for preventing the borehole from failure

(Designing the drilling fluid)

Elastic numerical analysis

Elastic analytical solution

Theoretical studies and modifications

Statistical and comparative studies

Comparison, validation and modification

Clarification of the assumed boundary conditions

A 3D failure criterion is selected through comparison, identification and modification

Figure 1.2 Demonstration of different phases in the stepwise research method adopted in this study


7

generalised Kirsch equations as an elastic analytical solution. Finite element analysis is also

carried out to calculate the induced stresses around the borehole based on newly proposed

boundary conditions, which more accurately represent the problem in the physical sense. The

current analytical solution will then be modified based on the finite element analysis,

incorporating the proposed boundary conditions.

Step B Five failure criteria, especially developed for rock material, are selected and

theoretically studied based on the fundamentals of the failure theory. A new three-dimensional

failure criterion is also developed by modifying the simplified Priest criterion. True-triaxial

tests are conducted and the results of the experiments along with nine sets of published true-

triaxial test data are applied to validate and compare the three-dimensional rock failure criteria

against the experimental data. As a result one failure criterion from among the five selected

criteria is identified as the best three-dimensional model for predicting the rock failure in

three-dimensional stress state.

Step C Following the stepwise analysis, the instability of a deviated borehole is predicted in

a case study. For this purpose, the maximum in situ stress induced around the borehole is

compared to the predicted failure stress of the rock material surrounding the borehole to

predict the initiation of failure at the borehole wall. In addition, to maintain safe drilling of an

inclined borehole and to prevent the rock material at the borehole wall from compressive and

tensile (hydro fracturing) failure, the minimum and maximum allowable mud weight are

determined

Examination of the influence of all factors affecting borehole stability, such as time,

temperature and mud chemistry, is beyond the scope of this research. Furthermore, since the

focus of this research is on borehole stability in good quality brittle rock material or rock

masses, for which the elastic solution is still valid (Brady and Brown, 1993), the failure is

assumed to take place when the elastic limit is reached. The progression of failure due to the

accumulation of micro-cracks and also post failure behaviour of rock are beyond the scope of

this thesis.


8

1.4. Organisation of the Thesis

This thesis is made up of two main sections consisting of six chapters overall, including this

introduction and the conclusion in Chapter 6. The first section, presented in Chapter 3,

discusses elastic stress analysis around vertical and deviated boreholes by means of analytical

and numerical approaches. The second section in Chapter 4 discusses procedures for

determining the failure stress of rock in the three-dimensional stress state.

The elastic analytical solution for estimating stresses around a borehole, which has been

drilled in an isotropic, homogeneous and linearly elastic rock was developed in 1962 by

Hiramatsu and Oka, and has been widely used in industry ever since. The applications of the

existing elastic analytical solution, which is also known as the generalised Kirsch equations,

are outlined in Cheaper 2. Moreover, the current techniques for evaluating the strength of rock

material in a three-dimensional stress state have evolved over the past few decades. In Chapter

2 the evolution and shortcomings of each of these methods are discussed.

Finite element analysis was carried out to create a numerical counterpart to the current

analytical solution for stress analysis around a borehole, aiming to eliminate the ambiguity

associated with the boundary conditions assumed for deriving the current analytical model.

The detailed procedure of the numerical analysis and the results of the cross-evaluation of the

analytical and the numerical models are presented in Chapter 3. Furthermore, a new set of

boundary conditions with respect to the physics of the problem is introduced in Chapter 3. The

number of simplifying assumptions is reduced and the current analytical model is modified to

bring it closer to reality. The analytical and numerical models are compared quantitatively in

Appendix B.

Current failure models, which have been especially developed for predicting rock failure in a

three-dimensional stress state, are discussed in detail in Chapter 4. A new three-dimensional

failure criterion is also proposed in Chapter 4, by modifying the simplified Priest criterion

introduced by Priest (2005). In order to compare and validate the three-dimensional rock

failure criteria, a number of true-triaxial tests were carried out at the University of Adelaide.

Furthermore, nine sets of published true-triaxial test data are applied to evaluate the accuracy


9

of the newly proposed rock failure criterion, in comparison with other existing three-

dimensional failure models. True-triaxial data sets used in this study are given in Appendix C.

In addition, an illustrative comparison between three-dimensional rock failure criteria and the

experimental data is given in Appendix D by plotting the major principal stress versus the

intermediate principal stress at failure.

Chapter 5 presents a case study in which the borehole instability is predicted by comparing the

induced major principal stress at the borehole wall to the predicted rock failure stress. The

major in situ principal stress around the borehole is calculated by means of finite element

analysis, based on the assumption of a new set of boundary conditions. The rock failure stress

under a three-dimensional stress state at the borehole wall is calculated by means of the

proposed three-dimensional rock failure criterion. These techniques are also applied in

Chapter 5 to calculate the minimum and maximum allowable mud weight for safely drilling a

deviated borehole. Chapter 6 presents conclusions and recommendations for future studies.

CHAPTER 2

Literature review

Page

2.1. Introduction 10

2.2. In situ stresses before the introduction of the borehole 10

2.3. Stress Analysis around the Borehole 12

2.4. Strength analysis of intact rock 16


10

2.1. Introduction

Underground formations are inherently subjected to a stress field, mainly due to the weight of

the overburden geo-materials and tectonic activities. Due to the introduction of a borehole to

the underground formations, the pre-existing stress field is redistributed which leads to stress

concentration at the borehole wall. The stability of the borehole depends upon whether the

formation surrounding the borehole fails or remains stable under the new induced stress field.

The answer to this question is of paramount importance for the stability evaluation of a

borehole.

Considering a borehole which has been drilled in a good quality rock mass, the following

information is required in order to evaluate the stability of the borehole:

- estimation or measurement of the pre-existing in situ stress field

- an accurate estimation of the induced in situ stresses in the borehole proximity

- an accurate estimation of the strength of the rock surrounding the borehole

2.2. In situ Stresses Prior to the Introduction of the Borehole

It is important to have an accurate estimation of the pre-existing state of stress in the

subsurface formations before a borehole is drilled since the stress boundary conditions for

modelling and estimating induced stresses around the borehole are determined based on the

pristine state of the in situ stress field. In general, the pre-existing stress state is a function of

depth; however, the manner in which the three principal stresses and their associated

directions vary with depth does not always follow a predictable pattern (Amadei and

Stephansson, 1997). These stresses will be influenced by topography, tectonic forces,

constitutive behaviour of the rock material and by the local geological history.

At any given depth below the Earth’s surface the stress state can be described by three

components; a vertical component, �,, due to the weight of the over lying rock at the depth Z,

equal to -., where - is the average unit weight of the rock (e.g. in KN/m3); and two horizontal


11

components such as �/, the major, and �0, the minor horizontal stresses. A key assumption

here is that these stresses are principal stresses, i.e. no shear stresses exist.

Many expressions for the variations of the magnitude of the vertical and horizontal in situ

stresses with depth, at specific sites or for different regions of the world, have been introduced

in the literature. Examples of stress profiles and stress variations are given by Brown and

Hoek (1978), Haimson and Lee (1980), Zoback and Zoback (1980), Zoback and Healy (1992),

Lim and Lee (1995) and Jaeger et al. (2007). However, detailed investigation of models and

expressions for estimating stress variations and stress profiles is beyond the scope of this

research.

It is important to note that, to date, no rigorous methods are available for accurately predicting

the situ stresses. Furthermore, the process of estimating in situ stresses should not be

considered as a substitute for their measurement (Amadei and Stephansson, 1997). Various

techniques for rock stress measurement were comprehensively explained by Amadei and

Stephansson (1997) such as (a) hydraulic methods including: hydraulic fracturing (Fairhurst,

1964), sleeve fracturing (Stephansson, 1983), Hydraulic Test on Pre-existing Fracture (HTPF)

(Cornet, 1986); (b) relief methods including: surface relief methods, borehole relief methods

and relief of large rock volumes; (c) jacking methods; (d) strain recovery methods and (e)

borehole breakout methods.

It merits noting that stress in rock masses cannot be measured directly and methods for

measuring in situ stresses basically consist of disturbing the rock and analysing the response of

the rock associated with the disturbance. This analysis is often undertaken based on

assumptions about the constitutive behaviour of rock, which relates the rock strains to applied

stresses. However, it is important to remember that due to the complex nature of rocks and

rock masses, constitutive modelling of rock behaviour is not usually a straightforward matter.

As pointed out by Amadei and Stephansson (1997), in good to very good rock conditions,

where the rock is essentially linearly elastic, homogeneous and continuous, and between well-

defined geological boundaries, rock stresses can be determined with an error of 110% � 20%

for their magnitude and an error of 110% � 20% for their orientation. On the other hand, in

poor quality rocks, i.e. weathered, weak, soft and heavily fractured, the measurement of rock


12

stresses is extremely difficult. In such cases the success rate of stress measurements is usually

low.

2.3. Stress Analysis around the Borehole

The stress distribution around a circular hole in an infinite plate in one-dimensional tension

was published by Kirsch (1898). Kirsch equations can also be generalised to calculate stresses

around vertical and deviated boreholes with anisotropic far-field stresses. The general form of

the Kirsch equations can be given by calculating the induced stresses around a deviated

borehole around which there exists a general stress state.

If in situ stresses in a formation are principal stresses with respect to a defined Cartesian

coordinate system, as illustrated in Fig. 2.1, the general stress state which is induced around an

inclined borehole can be given by transforming the in situ principal stress components into a

local Cartesian coordinate system. This local coordinate system is defined by transforming the

global coordinate system in such a way that the Z-axis coincides with the axis of the inclined

borehole. A transform from (X, Y, Z) to (L, M, N) can be obtained in two operations; a rotation

3 around the Z-axis and a rotation 4 around the Y-axis, in a manner that the Z-axis coincides

with the N-axis, as shown in Fig. 2.1. Such rotations can be undertaken by calculating the

direction cosines between the axes in the X, Y, Z coordinates and the corresponding axes in L,

M, N coordinates (Fig. 2.1) and by applying the following relationship:

Y

X

Z

�0

�/

�,

L

N

M r

θ

z

Figure 2.1 Coordinate system for a deviated borehole [after Fjær et al. (2008)]


13

=

Z

Y

X

lll

lll

lll

N

M

L

ZNZMZL

YNYMYL

XNXMXL

(2.3.1)

Where direction cosines, 567 in Eq.2.3.1, and consequently the rotation matrix 89:, are defined

as follows:

[ ]

−−

=

=ββαβα

ααββαβα

cossinsinsincos

0cossin

sincossincoscos

ZNZMZL

YNYMYL

XNXMXL

lll

lll

lll

R (2.3.2)

Accordingly, the general stress state can be expressed by a general stress tensor which results

from transforming the principal stress tensor from the global coordinate system (X, Y, Z) into

the local coordinate system (L, M, N) in the following manner:

[ ] [ ]T

v

h

H

zzzyzx

yzyyyx

xzxyxx

RR

=

σσ

σ

σσσσσσσσσ

00

00

00

(2.3.3)

Where 89: is the rotation matrix, given by Eq.2.3.2, and 89:; is the transpose of the rotation

matrix. Furthermore, components �/ and �0, in Eq.2.3.3, represent in situ maximum and

minimum horizontal stresses, respectively, and the component �, is the vertical stress due to

the weight of the over lying geo-materials.

It is important to note that the general stress tensor in Eq. 2.3.3 represents the pristine stress

state, before the introduction of the borehole. Defining a cylindrical coordinate system by

measuring the angle � counter-clockwise from the L-axis and the radial distance r from the

centre of the borehole, as illustrated in Fig. 2.1, the induced stresses around the borehole as a

result of redistribution of the virgin stress field due to the influence of the borehole, can be

given by the following stress tensor:

[ ]

=

zzzrz

zr

rzrrr

ij

σσσσσσσσσ

σ

θ

θθθθ

θ (2.3.4)


14

For an unsupported borehole of radius a, components of the stress tensor in Eq. 2.3.4 are given

by a three-dimensional version of Kirsch equations, which can also be referred to as the

generalised Kirsch equations as follows:

θσθσσσσ

σ 2sin43

12cos43

12

12 2

2

4

4

2

2

4

4

2

2

−++

−+

−+

−

+=

r

a

r

a

r

a

r

a

r

axy

yxyxrr

θσθσσσσ

σθθ 2sin3

12cos3

12

12 4

4

4

4

2

2

+−

+

−−

+

+=

r

a

r

a

r

axy

yxyx

( ) θσνθσσνσσ 2sin42cos22

2

2

2

r

a

r

axyyxzzz −

−−=

+−

+

−−=

2

2

4

4 2312cos2sin

2 r

a

r

axy

yxr θσθ

σσσ θ

( )

+−=

2

2

1sincosr

axzyzz θσθσσθ

( )

−+=

2

2

1cossinr

axzyzrz θσθσσ

(2.3.5)

A detailed description of the derivation of Eqs.2.3.5 was provided by Bradley (1979), who

referred to a report by Fairhurst (1968). However, as was also reported by Peska and Zoback

(1995), the generalised Kirsch equations were first published by Hiramatsu and Oka (1962). It

is important to note that there exists a sign error in the expression of �� (Eqs. 2.3.5) in

Bradley’s paper, as was also reported by Fjær et al. (2008). This error can be observed

elsewhere in the literature, for instance in the first edition of the “Petroleum Related Rock

Mechanics” (Fjær et al., 1992), and several other works. However, correct expressions of these

equations are given by Fairhurst (1968) and Hiramatsu and Oka (1968).

Eqs. 2.3.5 are used in linear elastic analysis of borehole stability for calculating the induced

stresses around an unsupported borehole, which has been drilled into nonporous materials.

Due to the superposition principle, pore pressure effects may simply be added. The borehole


15

influence is given by terms in �<�and �<=, which vanishes rapidly with increasing r. The

solutions depend also on the angle θ (Fig. 2.1) indicating that the stresses vary with position

around the borehole. Generally the shear stresses are non-zero. Thus, ��, �� and �� are not

principal stresses for arbitrary orientations of the borehole. At the borehole wall, Eqs. 2.3.5

are reduced to:

0=rrσ

( ) ( ) θσθσσσσσθθ 2sin42cos2 xyyxyx −−−+=

( ) θσνθσσνσσ 2sin42cos2 xyyxzzz −−−=

0=θσ r

( )θσθσσθ sincos2 xzyzz −=

0=rzσ

(2.3.6)

According to Jaeger et al. (2007) and Fjær et al. (2008), Eqs. 2.3.5 were derived based on the

assumption of plain strain normal to the borehole axis. However, as explained by Fairhurst

(1968), the general stress problem was divided into two separate problems; a plane strain

problem for calculating induced stresses around the borehole by considering only far-field

normal and in-plane shear stresses and an anti-plane strain problem for estimating the

influence of far-field, out-of-plane shear stresses on the induced stresses around the borehole.

In the plane strain problem it is assumed that there exists no displacement along the borehole

axis and all displacements or deformations occur in planes perpendicular to the axis of the

borehole. On the other hand, in the anti-plane strain problem the only deformation is assumed

to be a constant deformation along the borehole axis.

Although the generalised Kirsch equations, given by Eqs. 2.3.5, have been presented in several

works in the literature, the boundary conditions assumed for deriving this analytical elastic

solution have been poorly explained. Thus, to eliminate the existing ambiguity about the

boundary conditions assumed for deriving the generalised Kirsch equations necessitates a


16

detailed investigation of the simplifying assumptions made in defining the associated

boundary conditions.

After calculating the components of the induced stress state in the borehole proximity (Eqs.

2.3.4), in order to evaluate the stability of the borehole, it is essential to develop accurate

knowledge of the strength of the rock material surrounding the borehole wall. If the rock

material at the borehole wall is strong enough to tolerate the induced stresses, rock failure does

not initiate; otherwise the rock will fail. The rock failure stress under a given state of stress can

be predicted by means of a mathematical model, which sources its input parameters from the

rock material characteristics. Since the borehole stability analysis in this study is carried out

by means of linear elastic solutions, either numerically or analytically, the strength of intact

rock under the three-dimensional stress state around the borehole is the focal point of interest

in rock strength analysis. Intact rock refers to the non-fractured blocks which occur between

structural discontinuities in a typical rock mass. Failure of intact rock can be classified as

brittle which implies a sudden reduction in strength when a limiting stress level is exceeded

(Hoek, 1983).

2.4. Strength Analysis of Intact Rock

The maximum stress that can be sustained by rock material under a given set of conditions is

usually referred to as failure stress, which can also be interpreted as a measure of rock

strength. If the induced stresses in the proximity of an excavation exceed the failure stress of

the rock material or rock mass, failure occurs and the excavation may not be able to fulfil the

function for which it was excavated. Hence, failure stress is a key parameter in the design of

underground excavations (Hoek and Brown, 1980) and needs to be estimated as accurately as

possible. Therefore, the experiments for conducting studies on rock deformation and rock

strength must be designed to simulate the natural conditions as closely as possible (Mogi,

1966). Moreover, numerous empirical and hypothetical models have been introduced for the

estimation, or rather prediction, of rock strength. Empirical criteria have been formulated

predominantly based on laboratory experiments on rock specimens under the stress conditions

simulating those encountered in situ. Hypothetical failure criteria are also dependent upon


17

laboratory tests to source the relevant input parameters such as rock material constants which

are characteristic of a particular rock type.

Conventional experiments used for the study of the mechanical characteristics of rocks are

triaxial tests, in which cylindrical rock specimens are subjected to uniform lateral, confining

pressure (�� ) and an axial stress ��. Conventional triaxial tests have been widely used

because of the equipment simplicity and convenient specimen preparation and testing

procedures. However, such tests allow to simulate a special case only where the intermediate

and the minor principal stresses, �� and ��, are equal. To simulate a general stress state, for

example at the borehole wall (�� ' �� ' ��), more complex equipment and sophisticated

testing procedures are needed. A true-triaxial apparatus, which enables to apply three

independent stresses on a prismatic or cubic rock specimen, can be employed to simulate the

in situ, general stress state. However, due to equipment complexity, complicated testing

procedures and difficulties in specimen preparation, a limited number of true-triaxial

experiments have been conducted so far. According to Mogi (1971b) a common difficulty in

true-triaxial rock testing is to achieve three different yet uniform stresses. Non-uniform

stresses can introduce substantial errors into the calculations of failure stresses done according

to the elasticity theory using a linear stress-strain relationship.

A model used for the prediction of rock strength is usually expressed as a failure criterion

which is a mathematical formulation of the stress components governing the occurrence of

failure. A number of predictive models have been introduced in the literature to describe

brittle rock failure, among which the Coulomb (1773), which is a hypothetical failure model,

and the Hoek-Brown (1980) which is an empirical failure model, have been widely accepted

and applied by rock mechanics practitioners and scientists. The Coulomb and the Hoek-Brown

criteria formulate rock failure under a special stress condition, in which the intermediate and

the minor principal stresses are equal (�� ' �� ). These criteria do not take into account

the influence of the intermediate principal stress, �� on rock strength as it grows beyond the

minor principal stress, ��.


18

Coulomb criterion

The Coulomb criterion was introduced by Coulomb in 1773, who suggested that rock failure

in compression occurs when the value of the shear stress on a hypothetical plane is sufficient

to overcome the natural cohesion of the rock and also the frictional force that opposes motion

along the hypothetical failure plane. The Coulomb criterion gives the shear stress at failure as:

ϕστ tannc+= (2.4.1)

Where �> is the effective normal stress acting on the shear plane, and and ? are cohesion

and the angle of internal friction, respectively. When the Coulomb criterion is written in terms

of the principal stresses, the major principal stress at failure is given by:

31 σσσ qc += (2.4.2)

Where �� and �� are the maximum and minimum principal stresses, respectively, and �� is the

uniaxial compressive strength of intact rock and is given by the following expression:

ϕϕσ

sin1cos2

−= c

c (2.4.3)

The parameter q, in Eq. 2.4.2, is also defined as:

ϕϕ

sin1sin1

−+=q (2.4.4)

In the three-dimensional stress space (��, ��, ��), the Coulomb criterion can be represented by

a failure surface which is a cone with a hexagonal cross section on a plane perpendicular to the

hydrostatic axis (�� ).

Hoek-Brown criterion

Hoek and Brown (1980) introduced their failure criterion for evaluating rock strength as an

important parameter for designing underground excavations in hard rocks. The Hoek-Brown

criterion was derived based on the results from the research on the brittle failure of intact rock

and on model studies of jointed rock mass behaviour. The Hoek-Brown criterion was initially


19

developed based on intact rock properties and then these properties were reduced by

introducing factors, which were representative of the characteristics of joints in a rock mass.

According to Hoek et al. (2002) the generalised Hoek-Brown criterion, which is an empirical

model and is based on observed rock behaviour, is expressed in terms of principal stresses as

follows:

a

cbc sm

++=

σσσσσ 3

31 (2.4.5)

Parameters AB, & and C are rock mass constants and can be estimated from the Geological

Strength Index (GSI) as follows:

−

= 28

100GSI

ib emm

255.09

100

>==

−

GSIifaandes

GSI

25200

65.00 <−== GSIifGSI

aands (2.4.6)

The parameter A6 is the Hoek-Brown constant parameter A for intact rock material and

depends on rock type and mineralogy. Geological Strength Index (GSI) varies from 0 for

highly fractured rocks to 100 for intact rocks. Broadly speaking, GSI depends on the degree of

interlocking of the rock blocks and on the surface quality of the discontinuities in the rock

mass (Hoek and Brown, 1997). Hoek et al. (2002) introduced a disturbance factor D, which

indicates the amount of disturbance caused by blast damage and stress relaxation. The

disturbance factor D, ranges from 0 for rock masses adjacent to machine excavated

underground openings, to between 0.7 and 1.0 for open pit mine slopes, depending on the

quality of blasting. Hoek et al. (2002) proposed the following revised empirical expressions

for AB, & and C:

−−

= D

GSI

ib emm 1428

100

−−

= D

GSI

es 39

100


20

65.0

3

20

15

−

−

−+= eea

GSI

(2.4.7)

Eqs. 2.4.7 have the benefit of covering the entire range of GSI values in a single group of

expressions. Furthermore, from Eqs. 2.4.6 and 2.4.7 it is obvious that for intact rock,

parameters AB, & and C are calculated as A6, 1 and 0.5, respectively. Therefore, the

generalised Hoek-Brown criterion takes the following form:

5.03

31 1

++=

cic mσσσσσ (2.4.8)

Eq. 2.4.8 is the original form of the Hoek-Brown criterion, which was first introduced in 1980.

The Hoek-Brown criterion is widely accepted, predominantly because it fits experimental data

reasonably well and its input data can be determined simply by measuring the uniaxial

compressive strength (for determining ��), from mineralogical investigations (for

determining A6) and structural properties of the rock i.e. GSI, (for determining parameters C

and &). It is also noteworthy that if a more accurate estimation of input parameter A6 is

required, conventional triaxial tests can be conducted to measure this input parameter. In this

thesis the intact-rock version of the Hoek-Brown criterion (Eq. 2.4.8) is adopted. However,

adopting the generalised version would ultimately allow the borehole stability evaluation

techniques discussed in this thesis to be applied for fractured rock masses when sufficient

experimental data is available.

It is important to remember that both the Hoek-Brown and the Coulomb criteria incorporate

only the major, �� and the minor, �� principal stresses in the rock failure model on the

assumption that the intermediate principal stress, �� does not have any influence on rock

strength. However, numerous studies have revealed that rock strength is substantially

influenced when the intermediate principal stress grows beyond the minor principal stress.


21

2.4.1. The influence of intermediate principal stress on rock failure stress

Experiments and observations that revealed the potential effect of the intermediate principal

stress, �� on brittle failure of rocks started with Böker (1915), who carried out biaxial

extension tests on Carrara marble. Böker first applied an axial stress to the rock specimen and

kept it constant, i.e. �� DE&FCEF, and then raised the confining pressure, i.e. �� , until

failure occurred. Von Kármán (1911) also compared the results of the biaxial extension tests

and conventional triaxial compression (�� ' �� ) tests on the same rock, and it was clear

that Carrara marble was stronger when �� equalled ��, at any level of �� tested. Similar results

were observed and presented by Handin et al. (1967), who conducted similar conventional

triaxial compression and biaxial extension tests on a limestone, a dolomite and glass. Hollow

cylinder experiments by Hoskins (1969) on trachyte also showed that �� has a significant

influence on rock strength. Other researchers also have conducted similar types of

experiments. Mogi (1966), however, argued that their experimental procedures were not

accurate enough, because of non-uniform stress distribution at the ends of the specimens in the

Von Karman triaxial cell, i.e. end effects.

In 1967 Mogi measured the failure stress and the fracture angle of Dunham dolomite,

Westerly granite and Solnhofen limestone in extension tests (�� ' ��) after nearly

eliminating the end effects. He compared the results from compression and extension tests and

noted that the effect of �� on rock strength and fracture angle is indisputable. However, Mogi

(1971a) pointed out that in order to investigate more closely the influence of the intermediate

principal stress on brittle failure of rock, a high-pressure true-triaxial apparatus was needed to

test hard rock specimens under three independently applied principal stresses �� ' �� ' ��.

The true-triaxial apparatus designed by Mogi (1971b) consisted of a pressure vessel that

accommodated a rectangular prismatic rock sample of size 15×15×30 mm. Two sets of

pistons were employed to apply intermediate, �� and major, �� principal stresses, and the

minor, �� principal stress was provided by the confining pressure in the vessel to ensure a

uniform distribution of ��. A minor principal stress of the magnitude of 800 MPa could be

applied. In order to prevent the hydraulic fluid from intruding into the rock specimen, silicon

rubber jacketing was applied to the specimen sides subjected to confining pressure. Mogi


22

minimised friction on the sample faces subjected to piston loading by applying lubricants such

as copper sheet jacketing and Teflon or thin rubber sheets between specimen faces and

pistons. Stress concentration at specimen ends was found to be greatly reduced by the high

confining pressure (Mogi, 1966).

Mogi (1971b) tested a number of carbonate and silicate rocks and plotted the results in the

form of �� at failure versus �� for different families of tests in which �� was kept constant.

According to Mogi’s experimental studies, failure strength increases steadily with the

magnitude of �� until a plateau is reached following which strength tends to decline as ��

approaches ��. The best-fitting curve to experimental data for any given �� is downward

concave, but the strength as �� grows closer to �� remains higher than that when �� . This

behaviour is also predicted by the theoretical model proposed by Wiebols and Cook (1968),

which will be discussed further in Section 2.4.2.

The same results were obtained by Chang and Haimson (2000) and Haimson and Chang

(2000) who studied the deformational and strength characteristics of specimens of KTB

amphibolite and Westerly granite. Since Mogi’s pioneering work on the design and fabrication

of a true-triaxial apparatus, a series of true-triaxial testing machines have been developed to

investigate the ��-dependency of the failure stress. Haimson (2006) has reviewed the research

conducted over the last 100 years, characterising and formulating the influence of the

intermediate principal stress, �� on brittle failure of rock. According to Haimson (2006), there

is conclusive evidence that the intermediate principal stress has substantial effect on rock

strength in brittle field.

Scanning electron microscopy observations of the failure process conducted by Chang and

Haimson (2000) revealed that micro-cracks develop mainly parallel to the ��-direction as the

intermediate stress grows beyond ��. The micromechanical processes leading to brittle

fracture under true-triaxial stress conditions begin with the dilatancy onset, when the

development of micro-cracks is sub-parallel to the major principal stress ��-direction. As ��

increases, micro-cracks grow and localise, creating a shear-band dipping in the ��-direction.

Upon brittle fracture the shear band fails, forming the eventual main fracture. Similar

observations were also reported by Crawford et al. (1995) based on the results from sandstone


23

specimens failed under true-triaxial stress. With respect to deformation, Chang and Haimson

(2000) established that for the same �� the onset of dilatancy increases significantly with the

magnitude of ��, similar to observations by Mogi (1971b) in Mizuho trachyte. Thus, the

intermediate principal stress appears to extend the elastic range of the stress-strain behaviour,

for a given ��, and therefore to retard the onset of the failure process, suggesting a

strengthening effect of ��.

Yield and failure

In engineering contexts, the terms ‘yield’ and ‘failure’ often cause confusion. In order to avoid

such confusion a clear definition of both terms seems to be necessary. As defined by Priest

and Hunt (2005), ‘yield’ refers to the stress state at which the rock starts to develop rapidly

accelerating inelastic deformations, and ‘failure’ is the stress level at which the rock

disintegrates due to the development of macroscopic fractures. Therefore, when yield is

reached, the rock may still be able to carry some extra load or fulfil its engineering function.

Since for brittle rocks the portion of inelastic deformation after the yield point and before the

failure point is small, i.e. less than 3% of permanent deformation before failure (Heard, 1960),

a specific point on the stress-strain curve that can be attributed to yield stress cannot be easily

identified. However, the stress at some small permanent strain, such as 0.2%, can crudely be

taken as the yield stress. On the other hand, for ductile materials, the yield point on the stress-

strain curve can be determined more easily and there exists a single well-established three-

dimensional criterion for yield, which originated with Von Mises (Nadai, 1950) as follows:

( ) ( ) ( ) coct =−+−+−= 213

232

2213

1 σσσσσστ (2.4.9)

Eq. 2.4.9 states that the yield point is reached when the distortional energy, represented by the

octahedral shear stress �� , is equal to a constant , which is a constant characteristic of a

particular material. However, this is not the case in brittle rock the strength of which varies

considerably with confining pressure.

Mogi (1971b) measured the yield stress of three carbonate rocks, namely, Solnhofen

Limestone, Dunham Dolomite and Yumaguchi Marble, and four silicate rocks such as Mizuho


24

trachyte, Orikabe monzonite, Inada granite and Manazuru andesite, under three-dimensional

stress state. He observed that yielding appeared clearly in all carbonates and also trachyte.

According to his experimental results, Mogi (1971b) concluded that the yield stress is

markedly affected by ��, but scarcely affected by ��. He also pointed out that the measured

yield stresses were correlated well by the following formulation:

( )321 σσστ ++= foct (2.4.10)

Plotting the octahedral shear strength, �� versus mean normal stress GHIJHKJHL� � �� M,

Mogi (1971b) found that Eq. 2.4.10 can be satisfactory as a yield criterion for rocks (Fig. 2.2).

He also pointed out that this criterion includes the Von Mises criterion as a special case with

slope of the curve equal to zero. The physical interpretation of the generalised Von Mises

criterion (Eq. 2.4.10) is that yielding will occur when the distortional strain energy reaches a

critical value. This critical energy is not constant, but monotonically increases with the

effective mean normal stress (Fig. 2.2).

Mogi (1971b) also found that at failure the shear faulting takes place on a plane parallel to the

direction of �� and therefore, there is no stress component associated with the intermediate

principal stress on the failure plane. Hence, the mean normal stress contributing to failure is

100

150

200

250

300

350

100 200 300 400 500 600

τ(o

ct) M

Pa

σ (oct) MPa

Figure 2.2 Mean octahedral shear stress, NOPQ vs. mean octahedral normal )OPQ at yield for Dunham dolomite [after Mogi (2007)]


25

not �� , but its two-dimensional representation, i.e. �R,� � HIJHL� . Accordingly, Mogi (1971b)

suggested that a failure criterion for rock can be expressed in the following form:

( )31 σστ += foct (2.4.11)

Where S is a monotonically increasing function. The physical interpretation of Eq. 2.4.11 is

that failure takes place when the distortional strain energy, which increases monotonically

with the effective mean stress (�R,�), reaches a critical value on the failure plane.

Experimental studies on KTB amphibolite by Chang and Haimson (2000) and Westerly

granite by Haimson and Chang (2000) also mirrored Mogi’s conclusion of a general

formulation for brittle ‘failure’ of rock.

It should be noted that ‘yielding’ does not occur on any definite slip plane with a definite

direction, therefore, the mean stress GHIJHKJHL� � �� M is taken as the effective mean normal

stress when formulating a ‘yield’ criterion. Substantial evidence suggests that the intermediate

principal stress, �� has a strengthening effect on rock. It can be inferred that the intermediate

principal stress causes the yield stress of the rock to increase. As a result of the increase in the

yield stress of rock the failure stress of rock also increases. It is important, however, to note

that after the yield point is reached, the intermediate principal stress does not influence the

rock failure. Therefore, rock failure criteria, which neglect the influence of ��, do not

appropriately reflect the mechanical behaviour of rock under a general stress state. Several

theoretical and empirical three-dimensional failure criteria, which incorporate �� in the rock

failure formulation, have been introduced over the past few decades. In the following section

for a number of selected three-dimensional failure criteria detailed discussions are presented.

2.4.2. Frictional criteria

Theoretical or hypothetical criteria are also referred to as ‘frictional criteria’ after Priest (2010)

who commented that these criteria source their input parameters from one or more of the

parameters of uniaxial compressive strength, cohesion ( ) and the coefficient, or the angle, of


26

internal friction (?). Some frictional failure criteria, which have been more commonly used in

the petroleum and mining studies, are briefly explained below.

Drucker-Prager criterion

Drucker and Prager (1952) proposed a mean-stress-dependent failure criterion, combining the

Coulomb and the Von Mises criteria. Drucker and Prager suggested their criteria as:

12/1

2 BJAJ += (2.4.12)

Parameters T and U are rock constants and by defining ��, �� and �� as major, intermediate

and minor principal stresses, respectively, the mean principal stress, V� is given by:

octJ σσσσ =++=3

3211 (2.4.13)

The mean shear stress V� at failure is given by:

( ) ( ) ( )6

213

232

221

2σσσσσσ −+−+−

=J (2.4.14)

A physical interpretation of the Drucker-Prager criterion is that failure occurs when the

octahedral shear stress G�� W3 2⁄ V��/�M exceeds a certain value that depends on the

octahedral normal stress, �� . Parameters T and U can be estimated from the Coulomb shear

strength parameters cohesion, and the angle of internal friction ?. The following values for

the parameters T and U represent an inscribed cone to the Coulomb failure surface in the

principal stress space (�1, �2 , �3):

( ) ( )ϕϕ

ϕϕ

sin33

sin2

sin33

cos6

+=

+= ii Band

cA (2.4.15)

The circumscribed Drucker-Prager can be derived by defining T and U as:

( ) ( )ϕϕ

ϕϕ

sin33

sin2

sin33

cos6

−=

−= cc Band

cA (2.4.16)


27

The cross section of the inscribed and circumscribed Drucker-Prager and the cross section of

the Coulomb criterion on the deviatoric plane are presented in Fig. 2.3. A deviatoric plane is a

plane perpendicular to the hydrostatic axis in the principal stress space (�1, �2 , �3). Detailed

explanations on the principal stress space and the deviatoric plane and associated concepts are

given in Chapter 4.

Modified Wiebols and Cook criterion

Wiebols and Cook (1968) proposed a failure criterion based on the additional energy stored

around Griffith cracks due to the sliding of crack surfaces over each other. They hypothesised

that in a homogeneous specimen of rock that can be regarded as an elastic material there exists

a large number of uniformly distributed and randomly oriented closed, plane cracks, the

dimensions of which fall within a limited range. When subjected to stress, a volume of such

material stores within itself strain energy, which can be divided into two parts. Firstly, there is

the strain energy which would be stored in the same volume of material when subjected to the

same stresses, in the absence of any cracks. Secondly, there is the additional strain energy due

to the presence of cracks.

When all three principal stresses applied to the material are compressive, the surfaces of any

closed crack can be subjected only to normal compressive stress, �> and shear stress, . Let

Coulomb CDP IDP

Deviatoric plane

Circumscribed Drucker-Prager

Inscribed Drucker-Prager

Coulomb

Figure 2.3 The cross section of (a) the Coulomb, (b) the circumscribed and (c) the inscribed Drucker-Prager on the deviatoric plane


28

the coefficient of sliding friction between the opposite surfaces of the cracks be a constant �.

Sliding between the opposite surfaces of a crack under the influence of the applied stress

occurs if || � ��> ' 0. The quantity "|| � ��>$ is defined as the ‘effective shear stress’ and

the strain energy per unit volume of the material stored around the cracks, as a result of the

sliding produced by this stress, is defined as the ‘effective shear strain energy’. The effective

shear stress on any crack depends on the magnitudes of the principal stresses and the

orientation of the crack relative to the directions of these stresses.

The effective shear strain energy depends on the magnitudes of the effective shear stresses on

each crack in a unit volume, the number of such cracks, and their size and shape. It is assumed

that each crack contributes to failure when it slides, and that the strength of rock is determined

by some maximum value of the effective shear strain energy. It then follows that the strength

of rock is a function of its properties, and also a function of the magnitude of each of the

principal stresses. The strain energy criterion proposed by Wiebols and Cook (1968) had a

major drawback as the coefficient of friction between two crack surfaces (�) could not be

determined through a standard experimental procedure.

Zhou (1994) introduced a three-dimensional model for compressive rock failure, which was an

extension of the Drucker-Prager criterion with similar features to the effective strain energy

criterion proposed by Wiebols and Cook (1968). The three-dimensional criterion introduced

by Zhou (1994) is also referred to as modified Wiebols and Cook. According to this criterion,

yield is predicted to occur when:

211

2/12 CJBJAJ ++= (2.4.17)

Where V� and V� are given by Eqs. 2.4.13 and 2.4.14 and parameters A, B, and C are

determined such that Eq. 2.4.17 is constrained by rock strengths at both triaxial and biaxial

compressions. In the triaxial stress state (�� ' �� ), rock strength is given by the

Coulomb criterion, i.e. �� [ \��, and in the biaxial state of stress (�� ' ��) rock

strength is given by �� ] [ \��, where ] is the biaxial plane strength of the rock, (Wiebols

and Cook, 1968), and is defined as:

( )ϕσ tan6.01+= ciG (2.4.18)


29

With respect to constraining conditions and by substituting the uniaxial rock strength (�� , �� 0) into Eq. 2.4.17 and introducing four intermediate parameters, such as

�̂, �̂, ^� and ^= as follows:

( ) cqGU σσ −−+= 131

( ) cqGU σσ −−+= 12 32

( ) cqGU σσ −++= 122 33

( ) cqU σσ 2134 ++= (2.4.19)

Parameters A, B and C are defined as:

+−−=

2

127

3

1

2 q

q

U

U

UC

( )32

31 4CU

q

qB −

+−=

933

2ccc Cs

Aσσσ

−−= (2.4.20)

Modified Lade criterion

A three-dimensional failure criterion in terms of stress invariants and three material

parameters, was introduced by Kim and Lade (1984). However, the criterion was initially

developed for soil and was later modified for concrete by Lade (1982). The material

parameters can be determined from any type of strength test, including the simplest possible,

such as the uniaxial compression or conventional triaxial compression tests. The Lade

criterion is given as:

11

3

31 27 η=

−

m

aP

I

I

I

(2.4.21)


30

Where _� � �� [ �� [ �� and _� � "��$. "��$. "��$ and A is a rock mass constant. The value

of _��/_� is 27 under the hydrostatic stress condition, i.e. �� . Furthermore,

parameters �� and A in the Lade criterion can be determined by plotting "_��/_� � 27$ vs.

"ab/_�$ at failure in a log-log diagram and locating the best fitting straight line. The intercept

of this line with the vertical line GcdeI

M =1 is the value of ��, and A is the slope of this line.

In order to apply Eq. 2.4.21 to rock material, Kim and Lade (1984) added a constant stress

(C. ab) to the principal stresses before substitution in Eq. 2.4.21 to incorporate the cohesion

and the tension which can be sustained by rock material, as follows:

aPa.11 += σσ

aPa.22 += σσ

aPa .33 += σσ (2.4.22)

Where C is a dimensionless parameter and ab is the atmospheric pressure and is in the same

units as ��,�� CEf ��. According to Kim and Lade (1984), the parameter C, which plays a

very important role in characterising the tensile strength of rock, can be determined through

triaxial tests.

Ewy (1999) developed a modified version of the Lade criterion by introducing a shift constant

with units of cohesion as g� to shift the stress axes to the tensile region. In addition, he

subtracted the pore pressure in order to handle effective stresses. The modified Lade criterion

is expressed as follows:

( ) η+=′′

273

31

I

I (2.4.23)

Where

( ) ( ) ( )0130120111 pSpSpSI −++−++−+=′ σσσ

( )( )( )0130120113 pSpSpSI −+−+−+=′ σσσ (2.4.24)


31

In Eqs. 2.4.24, the parameter hi is pore pressure, which can be excluded from the formulation

for non-porous rocks, and parameters g� and � are material constant which can be directly

derived from the Coulomb cohesion and the angle of friction ? through the following

expressions:

ϕtan1c

S =

( )ϕ

ϕϕηsin1

sin79tan2 2

−−= (2.4.25)

Under general stress conditions (�� j �� j ��), the modified Lade criterion, as well as the

original Lade criterion, predicts a strengthening effect due to increase of the intermediate

principal stress. This is followed by a slight reduction in the rock strength once �� becomes

‘too high’.

2.4.3. Hoek-Brown based criteria

Despite its widespread acceptance and application, the Hoek-Brown criterion has some

limitations. In the case of anisotropy, for instance, the Hoek-Brown Criterion should not be

used unless allowance is made for this anisotropy. The strength anisotropy can be exemplified

by a fault passing through a heavily jointed rock mass or a block of intact rock. The rock mass

or the rock material may be treated as an isotropic medium to which the Hoek-Brown failure

criterion is applicable, but the fault must be treated as an anisotropic weakness plane along

which slip can occur at a much lower stress level than that which would cause failure in the

rock mass or the rock material. Another important limitation is that the Hoek-Brown criterion

was developed based on the assumption that only major and minor principal stresses

contribute to rock failure and that intermediate principal stress does not have any influence.

In order to take into account the influence of the intermediate principal stress, significant

efforts have been made to modify the Hoek-Brown criterion or to introduce a three-

dimensional failure criterion based on the Hoek-Brown criterion. Input parameters of such a


32

three-dimensional criterion are the same as those of the Hoek-Brown model. The reason why

such attention has been given to the Hoek-Brown criterion was outlined by Priest (2005) as:

- The Hoek-Brown criterion has been developed specifically for rock materials and rock

masses.

- Input parameters for the Hoek-Brown criterion can be derived from uniaxial testing of

the rock material, mineralogical examinations and measurements of the rock mass

fracture characteristics.

- The Hoek-Brown criterion has been applied successfully to a wide range of intact and

fractured rock types over the past decades (Hoek and Brown, 1997).

Pan-Hudson criterion

Pan and Hudson (1988) proposed a modified three-dimensional version of the Hoek-Brown

criterion for predicting the strength of weak rock masses. Based on the assumption that

intermediate principal stress influences rock failure, especially in the case of weak rocks, the

Pan-Hudson criterion was developed and expressed in terms of invariants of the deviatoric

stress tensor as follows:

ciic

sI

mJmJ σσ

=−+32

33 122 (2.4.26)

If the major, intermediate and minor principal stresses are denoted as ��, �� and ��,

respectively, then _� � �� [ �� [ �� and V� is the second invariant of the deviatoric stress

tensor, or the mean shear stress at failure, and is given by Eq. 2.4.14. If the Pan-Hudson

criterion is rearranged and is written in terms of �� , the octahedral shear stress at failure, it

yields the following expression:

cioctioctc

sI

mm σττσ

=−+322

3

2

9 12 (2.4.27)

Where �� is given by Eq. 2.4.9 and eI� � HIJHKJHL

� .


33

Priest (2010), through a comparative study, demonstrated that the Pan-Hudson criterion

exhibits a substantially different pattern from that of other three-dimensional failure criteria,

when predicting the failure of intact rock (Fig. 2.4).

Furthermore, according to Zhang and Zhu (2007), for biaxial extension (�� ' ��) or

triaxial compression (�� ' �� ) stress states, the Pan-Hudson criterion does not reduce to

the form of the original Hoek-Brown criterion. Furthermore, comparison with experimental

data conducted on various rock types suggests that the Pan-Hudson criterion under-predicts

the rock strength in triaxial compression, but over-predicts the strength in biaxial extension

(Zhang, 2008, Zhang and Zhu, 2007). Reasons for the limitations of the Pan-Hudson criterion

will be addressed in detail in Chapter 4.

Generalised Priest criterion

Priest (2005) developed a three-dimensional criterion by combining the tow-dimensional

Hoek-Brown criterion with the Drucker-Prager criterion. The Hoek-Brown criterion was

adopted for its simplicity in acquiring input rock parameters and because it had already been

widely used in the mining industry. The Drucker-Prager criterion was adopted for its common

Modified wiebols and Cook

Modified Wiebols and Cook-FUCS

Modified Lade

Pan-Hudson

Generalised Priest

Circumscribed Drucker-Prager

Simplified Priest Generalised Zhang-Zhu

Coulomb

Intermediate principal stress, ��, MPa

Maj

or p

rinci

pal s

tres

s at

failu

re, �

� M

Pa

Figure 2.4 Relation between intermediate and major principal stresses at failure for eight different failure criteria for a rock mass subjected to a minor principal stress of 15 MPa, with a uniaxial compressive strength of 75 MPa, mi = 19 and GSI = 90 (Priest (2010))


34

application in borehole and wellbore stability in the petroleum industry. The generalised Priest

criterion initially required a numerical iteration to calculate the effective maximum principal

stress at failure. Melkoumian et al. (2009) developed an explicit solution for the generalised

Priest criterion. The term generalised implies that the parameter a and the dilation parameter m

in the Hoek-Brown criterion are not taken simply as 0.5 and 1, respectively, and are calculated

using either Eqs. 2.4.6 or Eqs. 2.4.7. An explicit expression for the generalised Priest criterion,

as addressed by Melkoumian et al. (2009), is given as follows:

)(3 3231 σσσσ +−+= PHBf (2.4.28)

Where �� is the major principal stress at failure and ��/k and a are calculated as:

a

c

HBbc s

mP

+

=

σσσ 3 (2.4.29)

( )F

FEEHB 22

232

232

3σσσσσ

−−−++=

m (2.4.30)

Parameters l and m are defined as:

ca

ai

CE

CmF

σ2

3 1

=

+= −

(2.4.31)

Where n is given by:

( )c

imsC

σσσ

232 ++= (2.4.32)

Further detail on the generalised Priest criterion and its application for prediction of intact rock

strength will be addressed in Chapter 4.

Simplified Priest criterion

Priest (2005) proposed another three-dimensional version of the Hoek-Brown criterion by

defining a weighting factor � ranging from 0 to 1.0. He then specified that when � is equal to


35

0, the intermediate principal stress, �� has no influence on failure and when � is equal to 1 the

minor principal stress,�� has no influence. Priest (2005) suggested that the weighting factor w

depends only on values of the minor principal stress, ��, and that it can be calculated by the

following expression for a range of sedimentary and metamorphic rocks:

βασ 3≈w (2.4.33)

Where �� is the minor principal stress and 3 o 4 o 0.15, for the rock types examined. After

defining the weighting factor w, the minor principal stress in the Hoek-Brown criterion can be

defined as follows:

( ) 323 1 σσσ wwHB −+= (2.4.34)

The next step is to substitute the value for ��/k from Eq. 2.4.34 into the Hoek-Brown

criterion, given by Eq. 2.4.5, to calculate ��/k. Therefore, the major principal stress calculated

by the Hoek-Brown criterion is given by:

a

cbcHBHB sm

++=

σσσσσ 3

31 (2.4.35)

Then the simplified Priest criterion is expressed as:

( )32311 2 σσσσσ +−+= HBHBf (2.4.36)

The term �� is the major principal stress at failure in the three-dimensional stress state, i.e.

when �� ' �� ' ��. Further explanation on the simplified Priest criterion is given in

Chapter4.

Generalised Zhang-Zhu criterion

Zhang and Zhu (2007) developed a three-dimensional version of the Hoek-Brown criterion

based on the assumption made by Mogi (1971b) that a failure criterion can be expressed as:

( )31 σστ += foct


36

Where the function S is defined as a monotonically increasing function. Zhang and Zhu

(2007), then proposed their criterion as:

cmboctboctc

smm σσττσ

=−+ 2,2

22

3

2

9 (2.4.37)

Where the parameter �R,� is defined as �R,� � HIJHL� . Zhang (2008) highlighted that the

Zhang-Zhu criterion could be simplified to the original form of the Hoek-Brown criterion for

biaxial extension (�� ' ��) and triaxial compression (�� ' �� ) states of stress.

Furthermore, the only difference between the Zhang-Zhu and the Pan-Hudson criteria is the

inclusion of the intermediate principal stress in the third term, in the right hand side of these

two criteria, given by Eq. 2.4.27 and Eq. 2.4.37 (where both are written in terms of �� ).

In fact, the Zhang-Zhu criterion implies an assumption that the intermediate principal stress

plays its strengthening role up to a certain point, after which the influence of �� on failure will

be eliminated. This assumption accords with the experimental studies on rock performance

under three-dimensional stress. Zhang (2008) modified the Zhang-Zhu (2007) criterion so that

it could be applicable to all those rock masses to which the generalised Hoek-Brown criterion

applies. The generalised Zhang-Zhu criterion is expressed as:

( ) cmboctb

a

octac

smm σσττ

σ=−

+

− 2,

/1

1/1 2

3

22

31 (2.4.38)

CHAPTER 3

Stress analysis around a borehole

Page


3.2. Stress analysis around a vertical borehole 40

3.3. Stress analysis around a deviated borehole 53

3.4. Numerical counterpart of the generalised Kirsch equations 60

3.5. A modification to the generalised Kirsch equations 70


37

3.1. Introduction

When a borehole is introduced into an already stressed isotropic elastic rock, the pre-existing

stress field in the vicinity of the borehole is redistributed. The primary driving factor that leads

to borehole instability is the magnitude and deviatoric nature of the in situ rock stress that

becomes concentrated in the rock adjacent to the borehole wall. Accurate knowledge of the

new, induced stress field due to the introduction of the borehole is vital for drilling stable

boreholes. The geological stresses can be measured by a variety of deformation measurement

techniques including hydro fracturing, inelastic strain recovery and borehole breakout

estimations. However, due to difficulties associated with in situ stress measurements, it is

desirable for the induced stresses in the vicinity of the borehole to be estimated accurately by

means of numerical or analytical models.

In two-dimensional analysis of axially symmetrical structures it is generally convenient to

represent the state of stress and strain in terms of polar coordinates "�, �$. The angle � (0 to

2π) gives the anti-clockwise angle of rotation from a reference axis, such as the X-axis in Fig.

3.1. The distance r gives the radial distance from the centre of the axially symmetrical

structure. In two dimensions the normal stress components are �� (radial stress) and ��

(tangential or circumferential stress) with the associated shear stress ��. Two-dimensional

polar coordinates can be extended to three-dimensional cylindrical coordinates by defining the

Z-axis to correspond to the axis of the axially symmetrical structure. In this case the additional

normal stress component is �� (axial stress) and the associated shear stresses are �� and ��.

Two-dimensional and three-dimensional strains can be specified in the same way. It is

important to remember that the actual orientations of the radial and tangential stress or strain

components will vary with location around the axially-symmetrical structure. Polar and

cylindrical coordinate systems are adopted because the stress and strain states around axially

symmetrical structures generally vary directly as functions of the parameters r and �. The

parameters r and θ therefore not only serve to specify the orientation and location of the

stress/strain components, but also appear in functions that define the values of the stress/strain

components themselves at the specified location.


38

For an isotropic, linearly elastic and homogeneous material, Kirsch (1898) calculated stress

components around a circular hole, when principal stresses were acting at infinity (Fig. 3.1).

The radial, tangential and shear stresses at an anti-clockwise angle θ, measured from the X-

axis, and a radial distance r from the centre of the hole are given by the following equations:

θσσσσ

σ 2cos3412

12 4

4

2

2

2

2

+−

−+

−

+=

r

a

r

a

r

a yxyxrr

θσσσσ

σθθ 2cos312

12 4

4

2

2

+

−−

+

+=

r

a

r

a yxyx

θσσ

σ θ 2sin3212 4

4

2

2

−+

−−=

r

a

r

ayxr (3.1.1)

In order to apply Eqs. 3.1.1 for stress estimation around a borehole, the rock into which the

borehole is drilled has to be assumed as continuous, homogeneous, isotropic and linearly

elastic (CHILE) material. However, a rock with such properties rarely exists in nature.

Furthermore, in borehole stability evaluation, especially in the presence of major

discontinuities at the prospective location of the borehole, questions arise concerning the

� C

��

��

��

)q

)+

r

s

Figure 3.1 Stresses on an element at a radial distance r from the centre of a circular hole with radius a, in polar coordinates.


39

validity of elastic stress analysis in the design process and the potential effect of the

discontinuity on the stability of the borehole wall.

However, it should be noted that the elastic solution calculates the induced stresses based on

the assumption that the pre-existing stress field has been redistributed due to the introduction

of a hole into a CHILE material. Therefore, the elastic solution for estimating the induced

stresses, coupled with a predictive model for rock strength (failure criterion) can be applied to

model the rock behaviour adjacent to the borehole wall. If the intact rock does not fail under

the induced stresses, then the possibility of slip along the major discontinuity should be

investigated, considering the stress components acting on the faces of the discontinuity. On the

other hand, if the induced stress around the borehole, calculated by means of the elastic

solution, causes failure to initiate within the rock material, the extent to which the failure will

progress needs to be determined and, if the failure appears to be too extensive, a suitable

support system must be designed and implemented to resist the collapse of the borehole.

Furthermore, the numerical techniques of stress estimation can also be given consideration as

alternative approaches to analytical solutions, especially in cases of complicated constitutive

relationships with respect to rock behaviour. Nevertheless, according to Brady and Brown

(1993), in some cases, an elastic analysis presents a valid basis for design in a discontinuous

rock mass and in others, provides a basis for judgement of the engineering significance of a

discontinuity.

The three-dimensional version of Kirsch equations for stress estimation in the proximity of

vertical and deviated boreholes was first derived by Fairhurst (1968) and has been widely

applied in the mining and petroleum industries ever since. Generally, three-dimensional

Kirsch equations are either applied to rock in situ stress measurements or to instability

prediction of any underground structure with a circular cross section, such as tunnels,

wellbores and boreholes. In rock in situ stress measurements, by means of borehole breakout

or hydraulic fracturing methods, stresses around the borehole are measured and then

subsequently substituted into the three-dimensional Kirsch equations to calculate the far field

stresses. A detailed explanation of stress measurement methods is beyond the scope of this

study and the reader is referred to Amadei and Stephansson (1997) for further explanation on

rock stress and its measurement.


40

A detailed mathematical derivation of the 3D Kirsch equations was presented by Fairhurst

(1968) and Jaeger et al. (2007). In addition, these equations have been applied as a closed

form solution for calculating the induced stresses around a borehole by Amadei and

Stephansson (1997), Zoback et al. (2003), Al-Ajmi (2006) and Fjær et al. (2008). However,

apart from the mathematical derivation, there is no clear explanation of the boundary

conditions on which these equations have been based. On the other hand, a thorough

understanding of these boundary conditions is essential for this analytical solution to be

confidently applied to stress analysis in the excavation proximity. In order to provide an

unambiguous explanation of boundary conditions involved in the derivation of three-

dimensional Kirsch equations, detailed investigations were conducted on the mathematical

procedure and the simplifying assumptions adopted for deriving the three-dimensional

equations from the original two-dimensional Kirsch equations.

A numerical counterpart of the three-dimensional Kirsch equations was also created to provide

a further opportunity to validate and explain the boundary conditions and simplifying

assumptions involved in the analytical model. Simplifying assumptions are usually made to

facilitate the derivation of a closed form solution to a complicated problem. In some cases

without these simplifying assumptions, deriving a closed form solution may be impossible.

However, such assumptions, especially in the case of incompatibility with the physics of the

real problem, may sacrifice the accuracy of the model. Therefore, a numerical model

counterpart to the three-dimensional Kirsch equations enables one to further improve the

accuracy of the model by eliminating some simplifying assumptions, without which the

analytical solution would be impossible to be formulated.

3.2. Stress Analysis around a Vertical Borehole

Stresses before drilling the borehole

Consider a block of rock, located at a great depth beneath the Earth’s surface, to which far-

field principal stresses ��, �� and �� are acting in the Cartesian r, s and Z coordinate

directions, respectively. Measuring the angle, �, counter-clockwise from the r direction, the

initial values of radial (��t), tangential (��t) and vertical (��t) stress components, and also


41

the associated shear stress components (��t, ��tCEf ��t ), can be expressed in a cylindrical

coordinate system as follows:

zzzyxyx

rr σσθσσσσ

σ =−

++

=00

2cos22

02cos22 000

==−

−+

= rzzyxyx σσθ

σσσσσ θθθ

θσσ

σ θ 2sin20

yxr

−−=

(3.2.1)

In order to model the pre-existing in situ stress field, a block of rock can be assumed to be

removed from its position under the ground and to be loaded with the same in situ stress

components from zero. Under such conditions the initial values of the stress components

acting on an element located at an arbitrary point in the model, as illustrated in Fig. 3.2, can be

expressed in a cylindrical coordinate system as follows:

)(2cos22 11 yxzzz

yxyxrr σσνσσθ

σσσσσ +−=

−+

+=

02cos22 111

==−

−+

= rzzyxyx σσθ

σσσσσ θθθ

θ

Z

X

Y r

��t

��t

��t ��t

��t

��t

��t

��t

��t

��

��

��

Figure 3.2 The model of the pre-stressed block of rock into which the borehole will be drilled


42

θσσ

σ θ 2sin21

yxr

−−= (3.2.2)

In Eqs. 3.2.2 the vertical normal stress component, ��t, was calculated regarding the

Poisson’s effect and the Poisson’s ratio, ν.

Stresses after drilling the borehole

After a borehole of radius C has been drilled into an isotropic elastic rock block, which was

loaded from zero by the same stress components as the pre-existing in situ, the radial, ��,

tangential, ��, and in-plane shear, �� stresses acting on an element of rock at a radial

distance � from the borehole centre can be calculated by applying the well known Kirsch

equations, given by Eqs. 3.1.1. However, it is important to remember that Eqs. 3.1.1 were

originally developed for calculating stresses around a hole in a thin plane in two dimensions.

On the other hand, since the axis of the borehole extends into the third dimension, i.e. the Z-

axis of a Cartesian or cylindrical coordinate system, stress components along the borehole axis

in the third dimension also need to be calculated.

In three-dimensional engineering problems, general stress states and the associated

deformations are represented by means of second order tensors with six distinct components.

However, in structures where one dimension is considerably greater than the other two

dimensions, the strain components associated with the extended dimension are assumed to be

constrained by nearby materials and are substantially smaller than the cross sectional strains.

Considering the fact that the length of a borehole along its axis is substantially greater than the

cross sectional dimensions on a plane perpendicular to the borehole axis, longitudinal

deformations can be assumed to be negligible due to the constraint imposed by nearby geo-

materials. Therefore, if the borehole axis is assumed to be coinciding with the Z-axis, all strain

components along the Z-axis ((��, (�� and (��) are assumed to be zero and deformations are

expected to occur only in planes perpendicular to the borehole axis, i.e. (X, Y)-planes. Such

simplifying assumption is referred to as plane strain conditions. Furthermore, the strain tensor

which describes the plane strain conditions is expressed as:


43

[ ]

=00000

yyyx

xyxxεεεε

ε (3.2.3)

The stress state corresponding to the plane strain conditions is given by the following stress

tensor:

[ ]

=

zz

yyyx

xyxx

σσσσσ

σ00

00

(3.2.4)

The component �� can be temporarily removed from the analysis to effectively reduce the

three-dimensional problem to a two-dimensional problem, dealing only with in-plane terms.

The stress component �� will then be determined in a manner to maintain the constraint of

zero longitudinal strain, i.e. (�� (�� (�� 0.

Therefore, when far-field in situ stresses are considered as principal stresses with respect to

the borehole orientation (Fig. 3.3), considering Eqs. 3.2.2 and the assumption of plane strain

the vertical stress component, ��, and the associated out-of-plane shear stresses, �� and ��,

can be calculated as follows:

� X Y

Z

r ��

��

)q

)u � vw)q [ )+x

)+ ��

��

��

��

Figure 3.3 Demonstrating the conditions for applying plane strain assumption for calculating longitudinal stress components around a borehole


44

( ) ( ) θσσνσσσνσσ θθ 2cos22

2

1

−−=++=

r

ayxzrrzzzz

0== zrz θσσ (3.2.5)

Changes in the initial stress state due to the introduction of the borehole

The pre-existing in situ stress field is redistributed due to the introduction of the borehole.

Changes in the initial stress state due to the stress redistribution can be manifested as the

difference between the values of pristine stress components acting on a rock element before

drilling the borehole, as illustrated in Fig. 3.2, and stress components on the same element

after the completion of the drilling operations. Therefore, considering Eqs. 3.1.1, 3.2.1 and

3.2.5, changes in the pre-existing in situ stress field can be formulated as follows:

−

−+

+−=−=∆

2

2

2

2

2cos34220

r

a

r

ayxyxrrrrrr θ

σσσσσσσ

−−

+=−=∆

2

2

2

2

2cos3220

r

a

r

ayxyx θσσσσ

σσσ θθθθθθ

θσσ

σσσ θθθ 2sin232 2

2

4

4

0

−

+=−=∆

r

a

r

ayxrrr

( ) θσσνσσσ 2cos22

2

0

−−=−=∆

r

ayxzzzzzz

0=∆=∆ zrz θσσ (3.2.6)

Total induced in situ stresses

Total induced in situ stress components acting on the rock material adjacent to the borehole

wall can be calculated by adding the stress changes, given by Eqs. 3.2.6 to the initial in situ

stress components given by Eqs. 3.2.1. Therefore, total induced stresses in the vicinity of a

vertical borehole are given as follows:


45

θσσσσ

σσσ 2cos3412

12 2

2

2

2

2

2

0)(

+−

−+

−

+=∆+=

r

a

r

a

r

a yxyxrrrrrr t

θσσσσ

σσσ θθθθθθ 2cos312

12 4

4

2

2

0)(

+

−−

+

+=∆+=

r

a

r

a yxyxt

( )θ

σσσσσ θθθ 2sin321

2 4

4

2

2

0)(

−+

−−=∆= +

r

a

r

ayxrrr t

( ) θσσνσσσσ 2cos22

2

0)(

−−=∆+=

r

ayxzzzzzzz t

0)()(

==tt zrz θσσ (3.2.6)

After the completion of the drilling operations in a formation under pre-existing in situ

stresses, total induced stresses acting on an element at a radial distance r from the borehole

centre are given by Eqs. 3.2.6, provided that the far field in situ stresses are principal stresses

with respect to the borehole orientation, as was the case for the vertical borehole illustrated in

Fig. 3.3. Furthermore, Eqs. 3.2.6 show that �� takes its smallest values and �� takes its

largest values, i.e. �� and �� are at their most deviatoric state, at � � C. This observation is

important because rock failure will always be initiated at the point where the stresses are at

their most deviatoric state, which is the case for rock material adjacent to the borehole wall. It

is also important to note that Eqs. 3.2.6 have been derived based on the assumption that the

hole has been drilled into a pre-stressed block of rock, as illustrated in Fig. 3.3, and the

problem is entirely different from drilling a hole into a block and then loading the block from

zero. It is of significant importance to take into account this boundary condition when

developing a model for estimating the induced stresses around a borehole; nevertheless, it has

always been neglected and has not been clearly explained in the existing literature.

In order to validate the assumed boundary conditions in the analytical model, a finite element

analysis (FEA) was conducted using the commercial software package ABAQUS/6.9. This

FEA was used to numerically calculate the induced stresses in the vicinity of the borehole with


46

exactly the same boundary conditions as those assumed for developing the analytical model.

Some basic concepts of the finite element method applied in this analysis are presented in

Appendix A.

3.2.1. Numerical model of a vertical borehole

Results of stress measurement in Australia indicate that the vertical stress �, increases linearly

with depth at a rate of approximately 0.022 MPa/m, minor horizontal stress �0 at a rate of

approximately 0.015 MPa/m, major horizontal stress �/ at a rate of between approximately

0.022 and 0.025 MPa/m, and pore fluid pressure at a rate of approximately 0.01 MPa/m

(Hillis and Reynolds, 2000). Therefore, at a depth of 3000 m, the total principal stresses could

typically be �, � 66 MPa, �0 � 45 MPa and �/ � 75 MPa with a pore fluid pressure of 30

MPa in porous rocks.

Finite Element Analysis (FEA) was carried out, using the ABAQUS/6.9 software package, for

estimating the induced stresses around a borehole at a depth of 3000 m. Here as well as for the

analytical model, plane strain boundary conditions were assumed, i.e. assuming zero

deformation along the axis of the vertical borehole. Furthermore, the borehole was assumed to

be drilled in a rock with a Poisson’s ratio of 0.35 and elastic moduli of 48 GPa. The borehole

radius was taken as 0.08 m. Furthermore, it is of paramount importance to note that the

borehole is assumed to be drilled in a block of rock which is already stressed. In order to

incorporate this pre-stress field in the finite element model two alternative approaches can be

adopted as follows:

- The model space can be created as a block without a hole on which a given set of far-

field stresses are acting. The location of the prospective hole must be specified in the

meshing step. The same as for the analytical model, all displacements along the Z-axis

are supposed to be suppressed. After redistribution of the stress field in the model space

and reaching the state of equilibrium, elements which have been ascribed to the location

of the borehole will be removed from the model space and then the induced stresses due

to the element removal will be measured. It should be noted that as a result of


47

suppressing the displacements along the Z-direction the term �w�� [ ��x will act in

the opposite direction of the vertical stress, ��, due to the Poisson’s effect and therefore,

will be subtracted from the pristine value of ��.

- An alternative and more convenient approach for incorporating the pre-existing stress

field is to create the model space as a block which contains the borehole under the given

set of far-field stresses and define a constant nodal displacement in the Z-direction. The

magnitude of this constant displacement is calculated as:

( )E

d yyxx σσν += (3.2.7)

It merits noting that since the stiffness matrix is known (defined in the material property

in ABAQUS/6.9), by introducing a nodal displacement, given by Eq. 3.2.7, the

associated stress field is calculated and incorporated in subsequent steps of the finite

element analysis.

The second approach has been adopted for conducting the finite element analysis (FEA) in the

current study. The result of the numerical analysis was then compared to the existing

analytical model (Eqs. 3.2.6). The zone of influence around the borehole was assumed to be a

circle with a radius of 0.48 m which is six times greater than the borehole radius. Thus, the

radial distance from the borehole wall and the border of the zone of influence was 0.4 m. This

zone was discretised by 40 concentric circles and each one of these concentric circles was

discretised into 120 elements. Therefore, the radial distance from the centre of the borehole to

the centroid of each element can be calculated as:

39....,,2,1,01.0105 3 =

+×+= −

mmar (3.2.8)

Where a is the borehole radius and the parameter A is a counter for concentric circles,

counting from 0 for the circle immediately adjacent to the borehole wall and 39 for the circle

adjacent to the border of the zone of influence. Furthermore, the angle �, measured counter-

clockwise from the X-axis, as illustrated in Fig. 3.4, indicates the angular location of the

centroid of each element around the borehole and is given by:


48

( ) 120...,,2,1132

3 =−+= nnθ (3.2.9)

The calculated values for � and � from Eqs. 3.2.8 and 3.2.9 were substituted into the analytical

model (Eqs. 3.2.6) and then the corresponding stress components in analytical model and the

numerical model were compared to one another. The results of the quantitative comparison

between the numerical and analytical models are presented in Appendix B, Table B.1

According to Eqs. 3.2.6, induced stresses around the borehole not only vary as the radial

distance from the borehole wall increases, for a constant �, but also alter with changing

angular position around the borehole, for a constant r. Changes in the induced stresses in the

finite element model as a function of angular position around the borehole have been

illustrated in Figs. 3.5 and 3.6 and have been compared graphically with the analytical

solution. Since errors in the numerical analysis in all cases are less than 2%, when compared

with the analytical model, it can be inferred that the finite element analysis for estimating

stresses around a borehole, which is drilled into an isotropic, homogeneous and linearly elastic

rock, is valid and has been conducted accurately. Furthermore, since the finite element

analysis clarifies the boundary conditions assumed for deriving the analytical model, for

θ X

Y

r

Figure 3.4 Radial distance from the borehole centre and angular position of a given element


49

which there exists no clear explanation in the literature, this study can also be considered as a

cross-validation, as it sheds light on the procedure of formulating the analytical solution.

As can be observed in Figs. 3.5 and 3.6, stress concentration occurs at two opposite points at

the borehole wall, i.e. � = 0° and � = 180°, and in the direction of the minimum horizontal

stress, �0. Apart from the stress concentration, stresses are found to be highly deviatoric at

these two angular positions (Figs. 3.5 and 3.6). Recalling that borehole instability occurs due

to rock failure at the borehole wall, and knowing that a highly deviatoric stress state is an

underlying factor which leads to rock failure, these angular positions, around the borehole, are

of particular interest. However, since the induced stresses around the borehole also vary with

radial distance from the borehole wall, it also merits investigating the changes in induced

stresses along radial direction at � = 0° or 180° to determine at what distance from the

borehole wall the induced stresses are at their maximum deviatoric state.

50

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σrr

(MP

a)

θ (Deg)

Numerical model Analytical model

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

180.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σθθ

(MP

a)

θ (Deg)


0

360 �0

270

180

90

�/

0

360 �0

270

180

90

�/

Figure 3.5 Comparison between numerical and analytical model for variation of induced radial ()��) and tangential ()��) stresses around the vertical borehole at � = 0.085 m

51

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σzz

(MP

a)

θ (Deg)


-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σrθ

(MP

a) θ (Deg)


0

360 �0

270

180

90

�/

0

360 �0

270

180

90

�/

Figure 3.6 Comparison between numerical and analytical model for variation of induced vertical ()uu) and in-plane shear ()��) stresses around the vertical borehole at � = 0.085 m


52

Fig. 3.7 illustrates the changes in induced radial, ��, tangential, ��, and vertical, ��, stresses

along the radial direction, r. As is obvious from Fig. 3.7, the maximum values of induced

stresses occur at the borehole wall, i.e. where r = 0.08 m (borehole radius), and more

importantly, the induced stress components are highly deviatoric at the borehole wall (r = 0.08

m), compared to other points located at further distance from the borehole wall (Fig.3.7).

Furthermore, the in-plane shear stress component, ��, is zero at the borehole wall and, as

illustrated in Fig. 3.8, it increases dramatically with increasing radial distance from the

borehole wall until it reaches a maximum value at a distance close to the borehole wall, in this

case at 0.05m, from the borehole wall. After reaching its peak value, at a small distance from

the borehole wall, the in plane-shear stress, ��, declines gradually until it reaches a plateau

(in this case 0.8 MPa).

Accordingly, in the case of the elastic solution, rock failure is anticipated to initiate at two

opposite points around the borehole, i.e. � = 0° and � = 180°, in the direction of the minimum

horizontal stress, �0, and at the borehole wall due to the stress concentration and the deviatoric

nature of the stress state. However, it should be noted that the occurrence of rock failure is

highly dependent upon the strength of the rock material surrounding the borehole. Techniques

for estimating the strength of rock material will be comprehensively investigated in Chapter 4.

0

20

40

60

80

100

120

140

160

180

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Nor

mal

str

esse

s (M

Pa)

Radial distance from the borehole wall (m)

σrr (MPa)-An σrr (MPa)-Num

σθθ (MPa)-An σθθ (MPa)-Num

σzz (MPa)-An σzz (MPa)-Num

Figure 3.7 Comparison between numerical and analytical model for variation of induced stresses along the radial direction r, at � = 0, for the vertical borehole

)�

0

270

180

90

360 )�


53

According to Eq. 3.2.6, and with respect to Figs. 3.7 and 3.8, the stress state at the two

opposite points located on the borehole circumference, i.e. � = 0° and � = 180°, for a vertical

borehole at the depth of 3000 m is given by the following stress tensor:

[ ]

=

=

97.8600

092.1790

000

aaaar

ar

rarrr

ij

σσσ

σσσ

σσσ

σ

θ

θθθθ

θ

(3.2.10)

3.3. Stress Analysis around a Deviated Borehole

In cases where the far-field in situ stresses are not principal stresses with respect to the

borehole orientation, for example, in the case of a deviated borehole, as illustrated in Fig. 3.9,

a general stress state exists at the borehole proximity. Considering a block of rock, the upper

and bottom faces of which are perpendicular to the borehole axis, all components of the

associated stress tensor can be identified.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

In-p

lane

she

ar s

tres

s, σ

rθ(M

Pa)


σrθ-Num

σrθ-An

Figure 3.8 Comparison between numerical and analytical model for variation of induced in-plane shear stress along the radial direction r, at � = 0, for a vertical borehole


54

Subscripts (g) and (l) in Fig. 3.9 represent the far-field stress components, measured with

respect to the global and local coordinates, respectively. Since the following analyses were

conducted considering a local coordinate system as a reference the Z-axis of which coincides

with the axis of the borehole, the subscript L will be eliminated in any successive nomination

of components of the general stress tensor. Therefore, the general stress tensor, which

describes the stress state in the vicinity of a deviated borehole can be given as:

[ ]

=

zzzyzx

yzyyyx

xzxyxx

ij

σσσ

σσσ

σσσ

σ (3.3.1)

Assuming the axis of the borehole to be the axis of elastic symmetry (isotropic case), induced

stresses around a deviated borehole, for which far-field stresses are not principal stresses, were

first analytically formulated by Hiramatsu and Oka (1962) and later explained in detail in a

report by Fairhurst (1968). The strategy for deriving the analytical solution was to divide the

general stress problem into two separate problems:

- a plane strain problem for calculating induced stresses around the deviated borehole due

only to far-field normal stresses (��, �� and ��) and in-plane shear stress acting on a

plane perpendicular to the borehole axis (�� )

��"�$ ��"�$

��"�$

��"�$

��"�$

��"�$ ��"�$

��"�$

��"�$

��"�$

��"�$

Figure 3.9 General stress state in the vicinity of an inclined borehole

��"�$


55

- an anti-plane strain problem for calculating induced stresses in the vicinity of the

inclined borehole merely due to out of plane shear stresses (�� and �� )

Consequently, adopting the superposition method the general stress problem is assumed to be

as the summation of the plane strain and anti-plane strain problems, as illustrated in Fig.

3.10.

Stress analysis for the case (a) in Fig. 3.10 is the same as for a vertical borehole, as explained

in Section 3.2. In order to incorporate the effect of far-field in-plane shear stresses (Fig. 3.10

(b)), the induced stresses around the borehole due to the far-field normal stresses can be

superposed by the induced stresses due to the pure far-field in-plane shear stresses. It merits

noting that in both cases (a) and (b) in Fig. 3.10 stresses around the borehole are calculated

based on the assumption of plane strain, i.e. (�� (�� (�� 0. The corresponding stress

tensor, which describes the stress state in the cases (a) and (b) in Fig. 3.10, is also expressed

by the plane strain stress tensor, given by Eq. 3.2.3.

3.3.1. Stresses at the borehole wall due to far-filed in-plane shear, )q+ and

normal stresses, )qq, )++ and )uu

Consider a plane perpendicular to the borehole axis on which shear stresses �� are

acting (Fig. 3.11). If this plane is rotated to find the principal directions and the associated

principal stresses, then Kirsch equations (Eqs. 3.1.1) can be applied to calculate the total

stresses around the hole. The rotation matrix, for rotating a two-dimensional Cartesian

coordinate system (X, Y), is defined as follows:

(a) (b) (c)

= + + ��

��

��

��

��

��

��

��

��

��

��

��

Figure 3.10 Corresponding stresses for (a) and (b) plain strain problem and (c) for anti-plane strain problem


56

−=

ββ

ββ

cossin

sincosR (3.3.2)

The angle 4 by which the plane and the coordinates X and Y have to be rotated to put the plane

in a position where it has its faces perpendicular to the principal directions X ′ and Y′ (Fig.

3.11), can be calculated from the following relationship:

=

−

−

ys

xsT

yx

xy

σ

σ

ββ

ββ

σ

σ

ββ

ββ

0

0

cossin

sincos

0

0

cossin

sincos (3.3.3)

Subscript s for normal stress components in Eq. 3.3.3 indicates that the magnitude of the

normal stresses �� and �� is equal to the magnitude of shear stresses �� .

From Eq. 3.3.3 the angle 4 is calculated as �= and �� , as illustrated in Fig. 3.11.

Furthermore, stress components acting on an element at a radial distance � from the borehole

centre and at an angular distance measured counter-clockwise from the X-axis can be

calculated in a cylindrical coordinate system by substituting �� and �� for �� and ��,

respectively, and �� G� � �=M for � in Eqs. 3.1.1. Therefore, stress components around the

borehole due only to far-field in-plane shear stresses are calculated as follows:

r

θ

X

Y ��

��

s� r�

��

X

Y

4 � �4

�́� � �� ́� � ��

r

Figure 3.11 Demonstrating the method adopted for calculating induced stresses around a borehole due to pure far-field shear stresses, acting on a plane perpendicular to the borehole axis


57

θσσ 2sin34

14

4

2

2

)(

+−=

r

a

r

axyrr FS

θσσθθ 2sin3

14

4

)(

+−=

r

axyFS

θσσ θ 2cos32

14

4

2

2

)(

−+=

r

a

r

axyr FS

0)()(

==FSFS zrz θσσ (3.3.4)

Consequently, on the assumption of plane strain, �bb"��$ can be calculated as:

( ) θσνσσνσ θθ 2sin42

2

)()()( r

axyrrzz FSFSFS

−=+= (3.3.5)

The subscript FS in Eqs. 3.3.4 and 3.3.5 indicates that the stress components around the

borehole have been calculated by merely considering the effect of far-field in-plane shear

stresses on the induced stress field around the borehole. After superposing Eqs. 3.2.6 by Eqs.

3.3.4 and 3.3.5, components of induced stresses around the borehole due to the far-field

normal and in-plane shear stresses are given as follows:

θσθσσσσ

σ 2sin43

12cos43

12

12 2

2

4

4

2

2

4

4

2

2

)(

−++

−+

−+

−

+=

r

a

r

a

r

a

r

a

r

axy

yxyxrr NS

θσθσσσσ

σθθ 2sin3

12cos3

12

12 4

4

4

4

2

2

)(

+−

+

−−

+

+=

r

a

r

a

r

axy

yxyxNS

+−

+

−−=

2

2

4

4 2312cos2sin

2)( r

a

r

axy

yxr NS

θσθσσ

σ θ

( ) θσνθσσνσσ 2sin42cos22

2

2

2

)( r

a

r

axyyxzzz NS

−

−−= (3.3.6)


58

The subscript NS in Eqs. 3.3.6 indicates that stresses at the borehole proximity have been

calculated due only to the far-field normal and in-plane shear stresses.

3.3.2. Stresses at the borehole wall due to longitudinal shear stresses

()qu � )uq) and ()+u � )u+)

The closed form solution for stress components around the borehole due to the far-field

longitudinal shear stresses acting parallel to the borehole axis, i.e. the case (c) in Fig. 3.10, was

derived by Fairhurst (1968) based on the assumption of anti-plane strain. As opposed to plane

strain boundary conditions, which allow only for deformations in planes perpendicular to the

borehole axis, in anti-plane strain deformations are assumed to take place only along the axis

of the borehole and no deformation is allowed in planes containing the cross section of the

borehole.

Considering a block of rock as illustrated in Fig. 3.10 (c), for infinitesimal displacements

the strain tensor associated with out-of-plane shear stresses can be defined as follows:

[ ]

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

=

0

00

00

z

uy

y

u

z

u

x

uy

u

z

ux

u

z

u

zxz

zy

zx

ε (3.3.7)

However, under anti-plane strain conditions displacements along the X and Y directions are

assumed to be zero. Considering rectangular Cartesian coordinates, the displacement field that

leads to a state of anti-plane strain is given as follows:

( )yxuu

uu

zz

yx

,

0

′=

== (3.3.8)


59

Where !� and !� are displacements along the X and Y directions respectively and ^� is the

displacement in the Z direction which is defined as a function of x and y. Therefore, the strain

tensor associated with anti-plane strain conditions is defined as:

[ ]

∂∂

∂∂

∂∂∂

∂

=−

0

00

00

y

u

x

uy

ux

u

zz

z

z

strainplaneantiε (3.3.9)

According to Eqs. 3.3.8 and 3.3.9 anti-plane strain deformations can be visualised, considering

an element of rock with dimensions of f#, f� and f�, as illustrated in Fig. 3.12. Furthermore,

for an isotropic and linear elastic material the stress tensor that results from a state of anti-

plane strain can be expressed as:

[ ]

∂∂

∂∂

∂∂∂

∂

=

=−

0

00

00

0

00

00

y

uG

x

uG

y

uG

x

uG

zz

z

z

zyzx

yz

xz

strainplaneantiσσ

σ

σ

σ (3.3.10)

Where G is the shear modulus of the material.

Accordingly, induced stress components around the borehole when only far-field longitudinal

stresses are involved and based on the assumption of anti-plane strain boundary conditions are

given as follows:

0)()()()(

====OSOSOSOS zzrrr σσσσ θθθ

( )

+−=

2

2

1sincos)( r

axzyzz OS

θσθσσθ

( )

−+=

2

2

1cossin)( r

axzyzrz OS

θσθσσ (3.3.11)


60

The subscript OS in Eqs. 3.3.11 indicates that the stress components around the borehole have

been calculated by considering the far-field out-of-plane shear stresses only.

Details of the mathematical procedure for deriving longitudinal stress components at the

borehole wall, �� and ��b, is beyond the scope of this research and the reader is referred to

Fairhurst (1968) for further information. Eqs. 3.3.6 along with Eqs. 3.3.11 are referred to as

the generalised Kirsch equations, which are being widely used for estimating stresses around

boreholes, circular tunnels and any other underground structures with a circular cross section

in the petroleum and mining industries.

In order to further clarify the boundary conditions assumed in deriving the generalised Kirsch

equations, a finite element analysis (FEA) was conducted by assuming the same boundary

conditions as those for the analytical model. The FEA for stress analysis around a deviated

borehole was undertaken considering the stress conditions in the Earth’s crustal formations of

Australia, as given in Section 3.2.1, at the depth of 3000 m. The results of the FEA were

compared with the analytical model, namely the generalised Kirsch equations.

3.4. Numerical Counterpart of the Generalised Kirsch Equations

Assuming the far-field in situ stresses in the Earth’s crustal rocks in Australia at the depth of

3000 m as �0 = 45 MPa, �/ = 75 MPa and �� = 66 MPa, the general stress state induced in the

vicinity of a deviated borehole is expressed by the general stress tensor given by Eq. 3.3.1. It

= +

�!�� f� �!�

�# f#

X

Y

Z

��

��

��

f�

f�

f#

Figure 3.12 Deformations associated with anti-plane strain boundary conditions


61

was also assumed that the trajectory of the borehole was manifested by the trend/plunge

system as 125/10. To specify the components of the general stress tensor necessitates

transforming the stress tensor associated with the far-field in situ stresses from the global

coordinate system into a local coordinate system, one coordinate of which coincides with the

borehole axis. The rotation matrix for performing the transformation can be derived by

calculating angles between the axes of the local coordinate system and their counterparts in the

global coordinate system. Furthermore, the angle ��, between two lines of trend/plunge 3�/4�

and 3,/4, can be found from the following expression:

( )[ ] [ ]vuvuvuuv ββββααθ sinsincoscoscoscos +−= (3.4.1)

The trend and plunge of the axes of the global coordinate system (X, Y, Z) are given as

follows:

Global coordinate system

Coordinates Trend (Deg) Plunge (Deg)

X-axis 0 0

Y-axis 90 0

Z-axis 0 90

On the other hand, the trend and plunge of the axes of a local coordinate system (L, M, N)

which has one of its axes coincide with the axis of the inclined borehole with trend and plunge

of 125/10 are given as:

Local coordinate system

Coordinates Trend (Deg) Plunge (Deg)

L-axis 125 -10

M-axis 215 0

N-axis 125 80

Therefore, with respect to Eq. 3.4.1 the rotation matrix is calculated as follows:

[ ]

−

−−

−−

=

9848.01422.00996.0

0000.05736.08192.0

1736.08067.05649.0

R (3.4.2)


62

Furthermore, the stress state at the depth of 3000 m, in the Earth’s crustal rock of Australia

and with respect to the global coordinate system is given as a principal stress tensor as

follows:

[ ]

=

6600

0750

0045

pijσ (3.4.3)

Therefore, components of the general stress tensor, which describes the general stress state

around the deviated borehole, are calculated by transforming the principal stress tensor, given

by Eq. 3.4.3, and using the transformation matrix, 89:, given by Eq. 3.4.2, as follows:

[ ] [ ] [ ] [ ]

−−−−−−

==9738.654476.21487.0

4476.28697.548812.13

1487.08812.131565.65T

pijgij RR σσ (3.4.5)

The finite element model was created for the deviated borehole using ABAQUS/ 6.9. The

borehole radius was given as 0.08 m and the borehole was assumed to be drilled in an

isotropic, homogeneous and linearly elastic material with elastic modulus of 48 GPa and

Poisson’s ratio of 0.35.

Boundary conditions assumed for performing the FEA to calculate induced stresses around the

deviated borehole were the same as those assumed for deriving the generalised Kirsch

equations. It merits noting that separating the general stress problem into two problems and

assuming plane strain boundary condition for one and anti-plane strain boundary condition for

the other are simplifying measures which make the derivation of analytical solution possible.

However, in reality it is impossible for a block of rock to undergo deformations, on the one

hand, on the assumption of plane strain conditions, which is manifested by zero out-of-plane

deformations ((�� (�� (�� 0), and on the other hand, on the assumption of anti-plane

strain conditions, which allows only for the out-of-plane deformations such that (�� j (�� j 0

and (�� 0. It is also impossible to perform these incompatible boundary conditions on a

single finite element model and therefore, as for the analytical model, two separate models

have been created; one for estimating the induced stresses due to the far-field normal and in


63

plane shear stresses and the other for modelling induced stresses around the deviated borehole

due to the far-field out-of-plane shear stresses.

As is inferred from Eqs. 3.3.6 the radial stress, ��, is at its smallest and the tangential

stress,��, is at its largest values at r = a. Furthermore, as was explained in Section 3.2.1, in

the case of a vertical borehole subjected to far-field normal stresses only, the tangential stress

is at its maximum value at � = 0° and � = 180°, compared to any other angular positions

around the borehole. Points located on these two opposite angular positions on the

circumference of the vertical borehole lied in the direction of the minimum horizontal stress

and were referred to as stress concentration points (Figs. 3.5 and 3.6). However, when a

borehole is drilled into a block of rock which is subjected to far-field normal and in-plane

shear stresses, the two opposite points of stress concentration do not lie in the direction of the

minimum horizontal stress, �0, as they do in the case of a vertical borehole (Figs. 3.13 and

3.14).

An illustrative comparison between the results of the numerical stress analysis, i.e. FEA, and

the results of the estimation of induced stresses around the deviated borehole by means of the

analytical model, i.e. the generalised Kirsch equations, is presented in Figs. 3.13, 3.14 and

3.15. As can be observed, the results of the finite element analysis comply with the results of

the analytical model, which is indicative of validity of the finite element model. Furthermore,

the finite element model further clarifies the assumed boundary conditions and simplifying

assumptions adopted for deriving the generalised Kirsch equations for calculation of induced

stresses in the proximity of a deviated borehole. A quantitative comparison and the associated

error analysis are given in Appendix B, Tables B.1-B.5

64

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σrr

(MP

a)

θ (Deg)


0

20

40

60

80

100

120

140

160

180

0 30 60 90 120 150 180 210 240 270 300 330 360

σθθ

(MP

a)

θ (Deg)


Figure 3.13 Comparison between numerical and analytical model for variation of induced radial ()��) and tangential ()��) stresses around the inclined borehole at � = 0.085 m

�� (a)

180

270

�� /

��

90

360 0

�� 0

��

(b)

180

270

�� /

��

90

360 0

�� 0

65

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σzz

(Mpa

)

θ (Deg)


-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σrθ

(MP

a)θ (Deg)


Figure 3.14 Comparison between numerical and analytical model for variation of induced vertical ()uu) and in-plane shear ()��) stresses around the borehole at � = 0.085 m

(a)

��

180

270

�� /

��

90

360 0

�� 0

(b) ��

180

270

�� /

��

90

360 0

�� 0

66

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0 30 60 90 120 150 180 210 240 270 300 330 360σrz

(MP

a)

θ (Deg)


-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

0 30 60 90 120 150 180 210 240 270 300 330 360

σθ

z(M

Pa) θ (Deg)


Figure 3.15 Comparison between numerical and analytical model for variation of induced longitudinal shear stresses )�u and )�u, around the inclined borehole at � = 0.085 m

0 180

90

270

360

��

��

��

(a)

0 180

90

270

360

��

��

��

(b)


67

The region of stress concentration is represented in red in the contour model in Figs. 3.13 and

3.14. It merits mentioning that only two distinct points in these stress concentration region, at

which �� is at its maximum value, are of particular interest. Furthermore, induced stresses

around the borehole are at their most deviatoric state at these two opposite points of stress

concentration. Therefore, it is of utmost importance to determine the angular position of the

two opposite points of stress concentration. As illustrated in Fig. 3.13 (b), the graph, which

represents changes in tangential stress �� as a function of angular position � around the

borehole, shows four optimum points where the slope of the tangent line to the graph is zero.

Therefore, there are four angular positions around the borehole where the value of the

tangential stress is either maximum or minimum.

To find these angular positions necessitates differentiating the function ��, given by Eqs.

3.3.6, with respect to � and equating the derivative to zero as follows:

( ) 02cos3

122sin3

14

4

4

4

=

+−

+−= θσθσσ

θσθθ

r

a

r

a

d

dxyyx (3.4.6)

Therefore, the angular position where the tangential stress is at its minimum or maximum is

given by:

yx

xy

σσσ

θ−

=2

2tan (3.4.7)

Eq. 3.4.7 yields four values for the angle � all of which satisfy Eq. 3.4.6. Recalling that the

periodicity of tan 2� is ��, four angular positions are determined as: �, � [ ��, � [ �, � [ ��

� .

Substituting these values for � into the ��-function (Eqs. 3.3.6), the maximum and minimum

values of �� and, more importantly, the angular positions associated with the maximum value

of �� can be identified.

In the case of the deviated borehole, which was considered as a case example for developing

the finite element model, four angular positions at which the ��-function is either at its

maximum or minimum values are determined by applying Eq. 3.4.7, as is presented in Table

3.1.


68

Fig. 3.16 illustrates changes in induced stresses at the angular position � = 55.166° as a

function of radial distance from the borehole wall, for both numerical and analytical models.

Induced stresses are at their most deviatoric state at the borehole wall, where the induced

stresses are either at their maximum or minimum values at a given angular position.

Therefore, the rock failure is envisaged to initiate at the borehole wall and at two opposite

angular position where stress concentration occurs, in this case at � = 55.166° and � =

235.66°. In order to predict the rock failure at the borehole wall and at the two stress

concentration points the common strategy is to determine the stress state at these two points at

the borehole wall and then investigate whether the rock material is strong enough to sustain

the induced stress state. With respect to Fig. 3.16 (and also error analysis Tables B.1-B.5 given

in Appendix B), the induced stress state at the two stress concentration points, i.e. � = 55.166°

and � = 235.66°, at the wall of the deviated borehole which has been studied as a case

example can be identified by the following stress tensor:

[ ]

−−=

=70.8602.30

02.324.1790

000

zzzzr

zr

rzrrr

ij

σσσσσσσσσ

σ

θ

θθθθ

θ (3.4.8)

θ (Deg) 55.166 145.166 235.166 325.166

σθθ max min max min

Table 3.1 Determining the angular position of the two points of stress concentration

0 180

360

270

90

� = 55.166°

69

0

20

40

60

80

100

120

140

160

180

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Nor

mal

str

esse

s (M

Pa)


σrr (MPa)-An σrr (MPa)-Numσrθ (MPa)-An σθθ (MPa)-Numσzz (MPa)-An σzz (MPa)-Num

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

She

ar s

tres

ses

(MP

a)


σrθ (MPa)-Num σrθ (MPa)-Anσrz (MPa)-Num σrz (MPa)-Anσθz (MPa)-Num σθz (MPa)-An

Figure 3.16 Comparison between numerical and analytical model for variation of induced stresses along the radial direction r, at � = 55.166°, for the inclined borehole

��

�� /

�� 0

��

0

180

90

270

360


70

3.5. A Modification to the Generalised Kirsch Equations

In the existing analytical model (generalised Kirsch), stresses around the borehole are

estimated, on one hand, based on the assumption of plane strain in which no axial deformation

is allowed and, on the other hand, based on the assumption of anti-plane strain in which no in-

plane deformation is allowed and the only deformation is supposed to take place along the axis

of the borehole. The plane strain boundary conditions were assumed in order to facilitate the

calculation of the induced stresses around a borehole considering only the far-field in situ

normal and in-plane shear stresses and the anti-plane strain boundary conditions were adopted

to estimate the effect of the longitudinal, out-of-plane shear stresses on induced stresses

around the borehole.

The underlying reason for separating the general stress problem to two problems; one on the

assumption of plane strain and the other on the assumption of anti-plane strain was to reduce a

three-dimensional problem to two two-dimensional problems for which deriving the analytical

solution was feasible. The plane strain and anti-plane strain conditions are simplifying

assumptions based on which analytical solutions can be derived as approximations to real

general problems, where the mechanics of the three-dimensional problem allows for making

such assumptions. Considering a block of rock at the depth of 3000 m, into which a borehole

has been drilled, as illustrated in Fig. 3.17, the axial dimension is considerably greater than the

cross sectional dimensions, so it can be assumed that the axial deformations are constrained by

nearby geo-materials and, therefore, are negligible compared with cross sectional

deformations. Hence, the physics of the problem allows for approximately calculating the

induced stresses around the borehole due to a far-field general stress state on the assumption

of plane strain. It should be noted that all out-of-plane deformations are assumed to be zero on

the assumption of plane strain, i.e. (�� (�� (�� 0. Furthermore, the corresponding out-

of-plane shear stresses in the stress tensor associated with the plane strain conditions are also

assumed to be zero, i.e. �� 0, and the vertical normal stress, ��, is determined in a

manner to restrain zero out-of-plane deformations. The strain and stress tensors associated

with the plan stain conditions are given by Eqs. 3.2.2 and 3.2.3.


71

Therefore, the assumption of anti-plane strain conditions, in order to incorporate the effect of

the far-field out-of-plane shear stresses on the induced stresses around the borehole, as

explained in Section 3.3.2, is contradictory to the plane strain assumption. However, it was

reasoned by Fairhurst (1968) that since out-of-plane shear stresses do not have any impact on

the in-plane induced stresses around the borehole, i.e. ��, �� and �� (Eqs. 3.3.11), the

induced out-of-plane stresses around the borehole can be calculated separately and on the

assumption of anti-plane strain, which assumes a constant deformation along the borehole axis

as a function of gradients of the axial displacement !� in the X and Y directions as follows:

constanty

u

x

ufu zz

z =∂

∂∂

∂= ),( (3.5.1)

Nevertheless, it merits noting that although the longitudinal shear stresses do not affect the

induced in-plane stresses around the borehole, the assumed boundary conditions can be highly

influential on the calculated values for the out-of-plane stresses, namely ��, �� and ��.

A more appropriate approach for estimating stresses around a borehole when a far-field

general stress state is involved can be given by solving the three-dimensional problem and by

assuming that deformations along the axis of the borehole is suppressed by nearby geo-

materials as illustrated in Fig. 3.17. Although deriving an analytical solution to this three-

dimensional problem may be difficult or even impossible, numerical methods such as FEA can

be employed to solve the problem.

Figure 3.17 A section of a borehole at the depth of 3000 m

!� � 0

!� � 0

3000 m


72

The general strain tensor for defining the three-dimensional deformations of a rock element

with dimensions of f#, f� and f�, in a Cartesian coordinate system can be expressed in terms

of the infinitesimal displacements as:

[ ]

∂∂

∂∂

+∂

∂∂

∂+∂

∂

∂∂+

∂∂

∂∂

∂∂

+∂

∂

∂∂+

∂∂

∂∂

+∂

∂∂

∂

=

z

u

z

u

y

u

z

u

x

u

y

u

z

u

y

u

x

u

y

u

x

u

z

u

x

u

y

u

x

u

zyzxz

zyyyx

zxyxx

ε (3.5.2)

However, since the assumption of zero deformation along the borehole axis is also indicative

of zero displacement, i.e. !� = 0, along the axis of the borehole, which is assumed to be

coinciding with the Z-axis of the Cartesian coordinate system, the deformation of the rock

element in the proximity of the borehole and at the depth of 3000 m is assumed to be

manifested by the following strain tensor:

[ ]

∂∂

∂∂

∂∂

∂∂

∂∂

+∂

∂

∂∂

∂∂

+∂

∂∂

∂

=

0z

u

z

u

z

u

y

u

x

u

y

u

z

u

x

u

y

u

x

u

yx

yyyx

xyxx

ε (3.5.3)

The stress tensor corresponding to this stain tensor can be given as follows:

[ ]

∂∂

∂∂

∂∂

∂∂

∂∂

+∂

∂

∂∂

∂∂

+∂

∂

∂∂

=

zzyx

yyyx

xyxx

z

uG

z

uG

z

uG

y

uE

x

u

y

uG

z

uG

x

u

y

uG

x

uE

σ

σ (3.5.4)


73

Where E is the elastic modulus and G is the shear modulus of the rock material. In Eq. 3.5.4

the vertical normal stress, ��, is supposed to be determined in order to satisfy the condition of

zero displacement along the axis of the borehole. Therefore, �� can be defined as a function

of gradients of the longitudinal displacements in the X, Y and Z directions as follows:

),,(z

u

y

u

x

uf zzz

zz ∂∂

∂∂

∂∂=σ (3.5.5)

Applying the proposed boundary conditions, given by the strain tensor in Eq. 3.5.3, the results

of the finite element analysis indicate that the normal and in-plane shear components of the

induced stress state around the deviated borehole remain unaltered, compared to their values

calculated by means of the generalised Kirsch equations (Appendix B, Table B.5). Therefore,

Eqs. 3.3.6 can be applied to calculate the stress components ��, ��, �� and ��. However,

longitudinal shear stresses �� and �� substantially change under the proposed boundary

conditions, as illustrated in Fig. 3.18.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 30 60 90 120 150 180 210 240 270 300 330 360

σrz

(MP

a)

θ (Deg)

Analytical model Numerical model

-6

-4

-2

0

2

4

6

0 30 60 90 120 150 180 210 240 270 300 330 360

σθz

(MP

a)

θ (Deg)

Analytical model Numerical model

Figure 3.18 Changes in longitudinal shear stresses around the borehole under the proposed boundary conditions


74

Based on the results of the finite element analysis, the formulation of the out-of-plane shear

stresses �� and �� can be modified as follows:

( )

+−=

2

2

1sincos2

1

r

axzyzz θσθσσθ

( )

−+=

2

2

1cossin2r

axzyzrz θσθσσ (3.5.6)

The angular position of the points with stress concentration around the borehole can be given

by Eq. 3.4.7 as � = 55.166°, which is also confirmed by the results of the finite element

analysis for the deviated borehole considered as an example for finite element analysis in the

current study. Furthermore, at � = 55.166°, a row of elements in the radial direction can be

selected (Fig. 3.19) in order to indicate changes of the longitudinal shear stressres as a

function of radial distance from theborehole wall. Therefore according to Fig. 3.19 the induced

stress components at the two points of stress concentration are given by the following stress

tensor:

[ ]

−−=

=70.8685.10

85.112.1790

000

zzzzr

zr

rzrrr

ij

σσσσσσσσσ

σ

θ

θθθθ

θ (3.5.7)

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 0.1 0.2 0.3 0.4

She

ar s

tres

ses

Distance from the borehole wall

σrθ (MPa)-Num σrθ (MPa)-Anσra (MPa)-Num σra (MPa)-Anσθa (MPa)-Num σθa (MPa)-An

Figure 3.19 Changes in longitudinal shear stresses under the proposed boundary conditions, along the radial direction from the borehole wall

270

90

180 3600

� = 55.166°


75

According to Eq.3.5.7 the only stress component at the borehole wall which changes

compared to the analytical solution, i.e. the generalised Kirsch equations, is ��. It also merits

noting that although the values of the stress component �� differ from the values calculated

by means of the generalised Kirsch equations, at farther radial distances it takes the value of

zero at the borehole wall.

CHAPTER 4

Rock strength analysis in three-dimensional stress

Page


4.2. Definition of general, principal and deviatoric stress tensors 77

4.3. Failure function in principal stress space 80

4.4. Failure functions in deviatoric stress space 82

4.5. Failure criteria on deviatoric and meridian planes 87

4.6. Failure criteria especially developed for rock material 91

4.7. Experimental evaluation of rock behaviour under three-dimensional

stress 109

4.8. True-triaxial experiments at The University of Adelaide 124


76

4.1. Introduction

Apart from the evaluation of stress conditions in the borehole proximity, an accurate

estimation of rock strength under a three-dimensional stress state is vital for predicting the

borehole instability. In other words, when drilling through rock or when considering the

stability of borehole in good quality, brittle rock, investigating the behaviour of intact rock in

three-dimensional stress regime is a key parameter in designing and drilling stable boreholes.

The strength evaluation of intact rock is even more important when considering stability of

boreholes with smaller cross sectional dimensions compared to discontinuity spacing and at

great depths.

According to Paterson and Wong (2005), the mechanical properties of intact rock material

such as failure strength, fracture angle and ductility, are function of stress state, temperature

and strain rate. Mechanical behaviour of rock under a given stress state which is induced and

imposed on rock due to the introduction of an opening in the Earth’s crustal formations is the

focal point of interest of this chapter. The effect of stress state on rock strength has been

comprehensively investigated under conventional triaxial compression "�₁ ' �₂ � �₃$ and

extension "�₁ � �₂ ' �₃$ stress states. However, as pointed out by Mogi (2007), due to the

complicated procedure of true-triaxial experiments rock behaviour in general stress state

"�� ' �� ' ��$ has not been studied adequately. On the other hand, to confidently predict the

borehole instability due to the rock failure at the borehole wall an accurate estimation of rock

strength under the induced three-dimensional stress state in the borehole vicinity is required.

Rock strength can be interpreted as the ultimate loading capacity of rock material in a given

stress state. The stress state at which failure occurs is often referred to as failure stress. Since

most rocks fail with an abrupt failure and plastic deformation is barely observed before

macroscopic disintegration, the failure stress in this study is considered as the stress state at

which rock material is disintegrated, as defined by Priest and Hunt (2005). The strength of

rock material surrounding an underground structure at the depth of interest can be determined

by simulating the in situ stress conditions in experimental studies. However, due to difficulties

with reproducing in situ conditions at great depth and complex experimental procedures of


77

three-dimensional rock testing, it is desirable to predict the rock failure stress by means of an

accurate predictive model. Such a predictive model is usually presented as a failure criterion,

which is either an empirical or analytical formulation.

A failure criterion for a rock is a mathematical expression that defines the stress state as a

combination of stress components which leads to rock failure. Such a criterion is usually

expressed in terms of the stress tensor and material properties of rock. It is important to

remember that the term ‘failure’ implies that the rock has completely disintegrated. It is,

however, possible for the rock to become unserviceable in an engineering sense, if substantial

inelastic deformations develop. In this context the term ‘yield criterion’ is more appropriate to

be adopted. Rock failure criteria can be developed fundamentally from the mechanical

analysis of an assumed failure mechanism, or can be developed empirically by modelling the

observed behaviour of rock during laboratory or site tests.

4.2. Definition of General, Principal and Deviatoric Stress Tensors

The general stress state which is imposed on a block of rock (Fig.4.1) can be expressed by

means of a second order tensor which is often referred to as stress tensorijσ :

=

333231

232221

131211

σσσσσσσσσ

σ ij (4.2.1)

If the rock block shown in Fig. 4.1 is rotated so that all shear stresses on all faces are

eliminated (Fig. 4.2), then the normal stresses acting on each face are known as principal

��

��

��

��

��

��

��

1

2

3

Figure 4.1 Compressive general stresses on a block of rock

��

��


78

stresses and the associated directions are referred to as principal directions. The stress tensor

which describes this stress state is known as principal stress tensor, ��67��, and can be written

as:

[ ]

=

33

22

11

00

00

00

σ

σ

σ

σpij (4.2.2)

It should be noted that although in Eq. 4.2.2 stress components in the general stress tensor (Eq.

4.2.1) have changed due to the rotation of the rock block and the coordinate system, the actual

stress state remains unaltered. Furthermore, there are certain invariants associated with every

tensor which are independent of the orientation of the coordinate system. For example, a

vector is a simple first order tensor and it is represented by three components in a three-

dimensional space. The magnitude of these components depend on the coordinate system

chosen to represent the vector, but the length of the vector is a scalar and is independent of the

orientation of the coordinate. Similarly, in association with every second order tensor, such as

the stress tensor, there exist three independent invariant quantities. Accordingly, the first (_�),

second (_�) and third (_�) invariants of the principal stress tensor are defined as follows:

3211 σσσ ++=I

1332212 σσσσσσ ++=I

3213 σσσ=I (4.2.3)

1

2

3 ��

��

Figure 4.2 Principal stresses on a block of rock

��


79

Furthermore, the hydrostatic stress tensor, ��67�/, associated with the principal stress tensor

(Eq. 4.2.2) is defined as follows:

[ ] ijHijI δσ31= (4.2.4)

Where 1I is the first invariant of the principal stress tensor and the term 31I is referred to as the

mean normal stress. The 3×3 matrix ijδ is known as the ‘kronecker delta’ and is defined as:

=

≠=

=100010001

0

)(1

jiif

sumNojiifijδ

(4.2.5)

When considering rock performance under a given stress state it is important to note that even

rocks with no shear strength will not fail or disintegrate under hydrostatic stress state "�₁ ��₂ � �₃$. Under high hydrostatic stress state, due to the closure of pre-existing voids and

micro-fractures in the rock body, rock becomes rather compact and consequently stronger.

Therefore, it can be inferred that, as opposed to metal materials, the failure stress of rock is

dependent upon the mean normal stress. However, the main factor which causes the rock to

fail is the deviation of stress state from the hydrostatic state of stress. Hence, to predict the

rock failure it is necessary to evaluate the deviatoric nature of the stress state acting on the

rock.

Considering the principal stress tensor in Eq. 4.2.2, the deviatoric stress tensor, ��67��, can be

written as:

[ ]

=

33

22

11

00

00

00

S

S

S

dijσ (4.2.6)

Where the diagonal components in Eq. 4.2.6 are defined as:

ijjjiiI

S δσ31−= (No sum) (4.2.7)


80

Furthermore, the first (V�), second (V�) and third (V�) invariants of the principal deviatoric

stress tensor are defined as follows: (for simplicity g��, g�� and g�� are written as g�, g�

and g�)

)(3211 aSSSJ ++=

( ) ( ) ( )[ ] )(61 2

132

322

212 bSSSSSSJ −+−+−=

)(3213 cSSSJ = (4.2.8)

It is also noteworthy that:

sumNoSS jjiijjii σσ −=− (4.2.9)

4.3. Failure Function in Principal Stress Space

In general stress state "�₁ ' �₂ ' �₃$ all possible combinations of the stress components,

which cause the rock to fail can be represented by means of a mathematical formulation

known as a failure criterion. For an isotropic and homogenous material in a uniform stress

regime, the failure criterion can be expressed in terms of a stress tensor, ��67�, which satisfies

the following relationship:

( ) 0=ijF σ (4.3.1)

However, models for predicting the rock failure stress are commonly expressed in terms of

principal stresses. Therefore, the function F in Eq. 4.3.1 can also be written in terms of

principal stresses as:

( ) 0,, 321 =σσσf

(4.3.2)

The function f in Eq. 4.3.2, can be interpreted as a surface in the principal stress-space

(��, ��, ��), as illustrated in Fig. 4.3. This surface is a geometrical representation of all failure


81

points in the stress space and is, accordingly, known as the failure surface. All stress points

inside the failure surface are stress states at which the rock does not fail and any point located

on the surface represents a failure stress point. Stress points outside of this surface are

theoretically meaningless. However, when experimental data lies outside of a failure surface,

it is inferred that the associated failure criterion underestimates the rock strength. On the other

hand, observing experimental data inside the failure surface indicates that the rock strength has

been overestimated. According to Mogi (2007), one of the most fundamental problems of rock

mechanics is the study of the shape of the failure surface for various rock types.

It also merits noting that the analysis of induced stresses adjacent to an excavation will usually

produce a general stress tensor (where the shear stresses are non-zero) expressed relative to a

local set of axes. On the other hand, failure functions expressed in terms of principal stresses

are applicable only when the stress state is manifested by the principal stress tensor. In order

to apply such failure functions (Eq. 4.3.2) for estimating rock strength, it is necessary to

transform the local general stress tensor into the principal stress tensor. However, recalling

that the invariants of a second order tensor are independent of the orientation of the coordinate

system, it is more convenient to express failure functions in terms of the invariants of the

principal stress tensor or the deviatoric principal stress tensor to effectively eliminate the

transformation operations of the stress tensor from the procedure of the rock strength analysis.

)�

)�

)�

Figure 4.3 Failure surface in the principal stress space


82

4.4. Failure functions in deviatoric stress space

The stress space can be defined using a Cartesian coordinate system, each axis of which

represents one of the three principal stresses. Any arbitrary point a in the principal stress space

is identified by three stress components and represents a unique stress state. Furthermore, the

position of the point P in the stress space can be addressed by a stress vector )�� "��, ��, ��$,

as illustrated in Fig. 4.4. The line �₁ � �₂ � �₃, which makes equal angles with the three

principal stress axes is called the ‘Stress-space diagonal’ or the ‘Hydrostatic axis’ (Fig. 4.4).

If 3, 4 and - are angles between the hydrostatic axis and axes ��, �� and ��, respectively, the

following relation holds between the direction cosines:

3

1coscoscos === γβα (4.4.1)

A plane perpendicular to the hydrostatic axis which also contains the point P is called the

principal stress-deviator plane or simply the ‘deviatoric plane’. A deviatoric plane which

contains the origin of the principal stress space is known as the π−Plane (Fig. 4.5).

4

��

��

��

- )��

a"��, ��, ��$

3

��

Figure 4.4 Hydrostatic axis and the stress vector )�� in the principal stress space


83

Defining the unit vector C � �√� (1, 1, 1) along the hydrostatic axis, the magnitude of the

projection of stress vector, )�� , on the stress-space diagonal (the line �� ), can be

calculated as:

( ) octIa σσσσσ 33

3

3

11321 ==++=⋅ rr

(4.4.2)

Where, _� is the first invariant of the principal stress tensor and �� is the octahedral mean

normal stress. On a certain deviatoric plane, given by _� � DE&FCEF, the distance between the

point P and the hydrostatic axis which can be given as the magnitude of the vector � , as

illustrated in Fig. 4.6, can be calculated as:

( )22 arrrrr ⋅−= σσ (4.4.3)

Substitution of Eq. 4.4.2 into Eq. 4.4.3 yields the following relationship:

( ) ( ) ( )[ ] 22

1

213

232

221 2

3

1Jr oct ==

−+−+−= τσσσσσσr (4.4.4)

The magnitude of the vector � , which originates from the hydrostatic axis and terminates at the

point P on the deviatoric plane (Fig. 4.5), indicates a factor by which the given stress state

deviates from the hydrostatic stress state. Furthermore, a given stress point in the principal

�� a

��

��

)��

π−Plan

¢��

��

Figure 4.5 Deviatoric and £-plane

Deviatoric Plane

Hydrostatic axis


84

stress space, which is a Cartesian coordinate system, can be represented in a cylindrical

coordinate system as well. Two coordinates of such a cylindrical coordinate system are the

hydrostatic axis and the vector � on the deviatoric plane. The third coordinate, which is the

angle �, is measured counter-clockwise form one axis of a Cartesian coordinate system on the

deviatoric plane.

Such a Cartesian coordinate system on the deviatoric plane can be defined by transforming the

principal stress coordinates ( ��, ��, ��) so that the ��-axis coincides with the hydrostatic axis.

The axes of the transformed Cartesian coordinate system are labelled as ��, �� and �� in

Fig. 4.6. Accordingly, any point in the (��, ��, ��)-space can be transformed to the

(��, ��, ��)-space using a transformation matrix, through the following relationship:

−−

−

=

3

2

1

3

2

1

3

3

3

3

3

36

6

6

62

6

62

20

2

2

σ

σ

σ

σ

σ

σ

d

d

d

(4.4.5)

��

Hydrostatic axis

C

��

O

Q

��

�� a

��

��

��

)��

Deviatoric Plane

Figure 4.6 Cartesian coordinate system on the deviatoric plane


85

After defining three orthogonal coordinates (��, �� and ��) on the deviatoric plane the

angle � can be measured counter-clockwise from the ��-axis. The relationship between the

cylindrical, "�, �, ��$ and Cartesian, (��, ��, ��), components of the point P, as illustrated in

Fig. 4.6, can also be established. Considering the point P in the cylindrical coordinate system

in Fig. 4.6, the length QC can be calculated from the following relationship:

( )333

3

3

3cos 1321

321I

OCQC =++=

++== σσσσσσγ (4.4.6)

Where - is the angle between the hydrostatic axis (OC in Fig. 4.6) and the ��-axis, and _� is

the first invariant of the principal stress tensor. To calculate, for example, the third component

of the point P in the principal stress space, i.e. ��, it is necessary to add the magnitude of the

projection of the vector � along the QC-direction to the length of the QC, given by Eq. 4.4.6.

Components of the vector � in �� and �� directions are � cos � and � sin �, respectively, as

shown in Fig. 4.7. Therefore, considering Eq. 4.4.5 the magnitude of the projection of the

vector � in QC direction, parallel to the ��-axis, can be calculated as follows:

θθ sin6

6cos

2

2rrrQC

−+

−=r

(4.4.7)

Substituting � � |� | from Eq. 4.4.4 into Eq. 4.4.7 the third component of the stress point P in

the principal stress space can be calculated as:

33

4sin

3

2 123

IJ+

+= πθσ (4.4.8)

The first and the second components of the point P in the principal stress space (��and ��) can

be calculated in the similar manner. Therefore, the relationship between the components of the

stress point P in the cylindrical coordinate system "�, �, ��$ and in the principal stress space

(��, ��, ��) can be established as follows:


86

3

3

4sin

sin3

2sin

3

2 12

3

2

1IJ

+

+

+

=

πθ

θ

πθ

σ

σ

σ

(4.4.9)

Since the component r of a stress point in the cylindrical system is indicative of deviation from

the hydrostatic stress state, it can be concluded that the cylindrical coordinate system is more

convenient for demonstrating the deviatoric nature of a stress point and can be, appropriately,

referred to as the deviatoric stress space. Furthermore, in the deviatoric stress space, with

respect to Eq. 4.4.2, any point on the ��-axis, or the hydrostatic axis, can be addressed as

√��

_� and considering Eq. 4.4.4 the magnitude of the r component for any stress point is given

as W2V�. Therefore, the coordinates of the deviatoric stress space can be identified also as

(W2V�, � , √�� _�), in which _� is the first invariant of the principal stress tensor and V� is the

second invariant of the deviatoric stress tensor. The angle �, known as the ‘lode angle’,

according to Zienkiewicz et al. (1972) can be expressed in terms of second and third invariant

of the stress deviator tensor, V� and V�, respectively.

Considering Eq. 4.4.5, the term W2V� sin �, which represents the magnitude of the projection

of the vector � in the ��-direction (Fig. 4.7) can also be written as:

Figure 4.7 Polar components of point P on the deviatoric Plane

�

P

Deviatoric Plane

��

��

W2V� sin �

W2V� cos �

|� | � W2V�


87

( )3212 26

6sin2 σσσθ −+−=J (4.4.10)

Recalling Eq. 4.2.5 and rearranging Eq. 4.4.10, sin � can be calculated as:

2

2

2

3sin

J

S=θ (4.4.11)

According to Eqs. 4.2.7 (c), considering the trigonometric identity "sin 3� � �4 sin� � [3 sin �$ and knowing that g� � �"g� [ g�$, the lode angle � can be expressed in terms of the

second (V�) and third (V�) invariants of the principal deviatoric stress tensor, as follows:

23

2

3

2

333sin

J

J−=θ (4.4.12)

Since the failure stress of rock, as a brittle material, depends upon the effective mean normal

stress G¨I� M and failure occurs only under highly deviatoric stress state, it is more convenient to

express rock failure criteria as a function of the first invariant of principal stress tensor, _�, and

the second and third invariants of the principal deviatoric stress tensor (V� and V�).

Accordingly, the rock failure function in deviatoric stress space (W2V�, � , √�� _�) can be

expressed as:

( ) 0,3sin, 12 =IJF θ (4.4.13)

Where &©E 3� is given by Eq. 4.4.12. It also merits noting that by employing Eq. 4.4.9, any

failure criteria in terms of principal stresses can be expressed in terms of invariants of the

principal and principal deviatoric stress tensors.

4.5. Failure Criteria on Deviatoric and Meridian Planes

The failure function expressed by Eq. 4.4.13, represents a failure surface in the deviatoric

stress space. Due to the presence of the term sin 3� in Eq. 4.4.13, a number of general

symmetry properties of the failure function can be addressed. The trace of this failure surface


88

on an arbitrary deviatoric plane is obtained for _� = constant. As the Sin-function is periodic

with a period of 360° it is straightforward to conclude that the failure function in Eq. 4.4.13 is

periodic with a period of 120° and therefore the trace of the failure surface on the deviatoric

plane is repeated in every 120° and the distance 9 � |� |, between hydrostatic axis and the

trace of the failure surface on the deviatoric plane, is the same for � and for � [ 120° as well

as for � [ 240° (see Fig. 4.8 (a)).

Due to the periodicity of 120° the cross sectional curve is also symmetric about � � 90°, � � 210° and � � 330°, as illustrated in Fig. 4.8 (b). Furthermore, setting � � 30° 1 4

yields sin"90 � 34$ � sin"90 [ 34$ and accordingly, the magnitude of the vector � is

identical for � � 30° [ 4 and � � 30° � 4, which indicates that the trace of the failure

surface on the deviatoric plane is also symmetric about � � 30° and thereby also symmetric

about � � 150° and � � 270°, (see Fig. 4.8 (c)). Similarly, for � � 90° 1 3, the magnitude

30

210

60

240

90

270

120

300

150

330

180 0

« «

"¬ [ ®$ "¬ � ®$

(d)

30

210

60

240

90

270

120

300

150

330

180 0

"� � ¯$

"� [ ¯$ «

«

(c)

30

210

60

240

90

270

120

300

150

330

180 0

(b)

«

« «

30

210

60

240

90

270

120

300

150

330

180 0

"�$

"� [ �°°$

« «

«

(a)

Figure 4.8 Symmetry properties of a failure criterion on the deviatoric plane

"� [ ��°$


89

of the vector � is the same and thereby the trace is symmetric about � � 90° , � � 210° and � � 330°, (see Fig. 4.8 (d)). The symmetry properties shown in Fig. 4.8 imply that the

trace of the failure surface on the deviatoric plane is completely characterized by its form for

�30° ± � ± 30° and that this form is repeated in other sectors of the deviatoric plane.

Furthermore, if �₁ ' �₂ ' �₃ are principal stresses, the intermediate principal stress can be

written as:

( ) 101 312 ≤≤+−= ασασασ (4.5.1)

Substituting Eq. 4.5.1 into Eq. 4.2.5 gives:

( )( )

( )( )

( )( )313

312

311

231

213

1

131

σσα

σσα

σσα

−−−=

−−=

−+=

S

S

S

(4.5.2)

Substituting g� from Eq. 4.5.2 into Eq. 4.4.11 yields:

12

21sin

2 +−

−=αα

αθ (4.5.3)

Since the parameter 3 ranges between 0 and 1 (0 < 3 < 1), it follows from Eq. 4.5.3 that the

angle � ranges from � �² and

�² (� �

² ± � ± �²). Therefore, with the ordering of the principal

stresses such that �₁ ' �₂ ' �₃, all stress states are covered by the angle � ranging between

� �² and �².

The ‘meridians’ of the failure surface are the curves where � � DE&FCEF applies. In other

words the ‘meridional’ curves are obtained by the intersection of the failure surface with a

plane containing the hydrostatic axis.


90

Accordingly, the meridians can be depicted in a "�, �$ coordinate system, known as the

‘meridional plane’ (Fig. 4.9). With respect to Eqs. 4.4.2 and 4.4.4, coordinates � and � are

defined as follows:

( )

2

1321

2

3

3

3

3

J

I

=

=++=

ρ

σσσξ (4.5.4)

For rock materials two meridians are of particular interest. When ��> �� = �� applies, then in

Eq. 4.5.1, 3 = 1 and from Eq. 4.5.2 � � � �². This meridian is termed the ‘compressive

meridian’, as the stress state ��> �� = �� corresponds to a hydrostatic stress state superposed

by a compressive stress in the ��-direction. This stress state is often referred to as triaxial

compression in rock mechanics experiments:

�� ' �� ©. ³. � � � �6 DAh�³&&©%³ A³�©f©CE

Uniaxial compressive stress state is located on the compressive meridian, and so is the triaxial

compressive stress state when the intermediate and the minor compressive principal stresses

are equal. When ��= �� > �� holds then Eq. 4.5.1 postulates that 3 = 0 and Eq. 4.5.2 calculates

the angle � as �². This meridian is termed the ‘tensile meridian’, as the stress state �� '�� corresponds to a hydrostatic stress state superposed by a tensile stress in the ��-direction.

This stress state is often referred to as triaxial extension in experimental studies of rock

mechanics:

�

Hydrostatic axis

��

��

��

¢��

� a"��, ��, ��$

Figure 4.9 Meridional plane (´ � µ coordinates) [after Ottosen and Ristimna(2005)]


91

�� ' �� ©. ³. � � �6 F³E&©5³ A³�©f©CE

The points where the tensile and compressive meridians intersect the deviatoric plane are

illustrated in Fig. 4.10.

4.6. Failure Criteria Especially Developed For Rock Material

In this section a group of empirical three-dimensional rock failure criteria is studied in detail.

Since the input parameters of these three-dimensional models are the same as those for the

Hoek-Brown criterion, they can also be referred to as three-dimensional Hoek-Brown based

criteria. The three-dimensional Hoek-Brown based criteria are first expressed in terms of the

invariants of the deviatoric stress tensor using the method outlined in Section 4.4. Next the

radial distance between the hydrostatic axis and the trace of the failure surface on the

deviatoric plane is calculated in a unified way and by assuming _�, the mean normal stress, as

constant. The three-dimensional failure surface in the principal stress space is then plotted by

reproducing the trace of the failure surface along the hydrostatic axis. For this purpose _�

needs to be defined as a variable.

30

210

60

240

90

270

120

300

150

330

180 0

��

Tensile Meridian

��

Compressive Meridian

��

Figure 4.10 Intersection of tensile and compressive meridians with the deviatoric plane


92

4.6.1. The Hoek-Brown criterion

According to Hoek and Brown (1980), the Hoek-Brown criterion for estimating the failure

stress of intact rock material can be expressed as a function of the major (��) and minor (��)

principal stresses as:

( ) ( ) 0, 2

1

233131 =+−−= cciHB smF σσσσσσσ (4.6.1)

Where the term A6 is the Hoek-Brown parameter m for intact rock and the parameter s for

intact rock is 1. Substituting the relevant formulation for �� and �� from Eqs. 4.4.9 into Eq.

4.6.1, the Hoek-Brown criterion can be expressed in terms of invariants of the principal

deviatoric stress tensor as follows:

( )

≤≤−

=

+++

−−=

66

03

cos3sin3

cos2 2122

21

21

πθπ

σσθθσθ cci

ciHB sImJ

mJF

(4.6.2)

Rearranging Eq. 4.6.2, the Hoek-Brown criterion for intact rock can be written as a quadratic

equation in terms of V� in the following form:

033

3sin2

cos4 122

22=−−

++

= s

ImJm

JF

c

i

c

i

cHB σσ

πθ

σθ (4.6.3)

According to Eqs. 4.4.4 and 4.6.3 and considering Fig. 4.5, the radial distance from the

hydrostatic axis to any point on the trace of the Hoek-Brown failure surface on a certain

deviatoric plane, given by _� � DE&FCEF, can be calculated as follows:

+++−== s

ImJr

c

iHBHB

cHB HB σ

ςλλςσ

34

2

22 12

2 (4.6.4)

Where parametersς and λ are defined as follows:


93

663

3sin2

cos42

πθππθ

λ

θς

≤≤−

+=

=

i

HB

m (4.6.5)

Furthermore, assuming _� as constant, Eqs. 4.6.4 and 4.6.5 can be applied to plot the cross

section of the Hoek-Brown failure surface on the deviatoric plane. As illustrated in Fig. 4.12,

the cross section of the Hoek-Brown failure surface on the deviatoric plane is a hexagon.

When �� , from Eqs. 4.5.1 and 4.5.3 it follows that � � � �². Substituting � as � �

² into

Eq. 4.6.4, the distance between the hydrostatic axis and sharp corners of the hexagonal cross

section of the Hoek-Brown criterion, �/k�, on the deviatoric plane, is calculated as follows:

+++−== s

ImmmJr

c

iiicHBHB ss σ

σ3

123

4

3

2

6

22 1

2

2 (4.6.6)

Likewise, �� indicates that in Eq. 4.5.1 3 � 0 and form Eq. 4.5.3 � is given as � ².

Therefore, the distance between the hydrostatic axis and blunt corners of the hexagonal cross

section of the Hoek-Brown criterion (�/kB), is calculated as follows (Fig. 4.11):

+++−== s

ImmmJr

c

iiicHBHB bb σ

σ3

12336

22 1

2

2 (4.6.7)

�� (MPa)

�� (MPa)

Deviatoric plane T

B

S

�/kB

�/k�

Figure 4. 11 The cross section of the Hoek-Brown failure surface on the deviatoric plane


94

Accordingly, sharp corners in the hexagonal cross section of the Hoek-Brown failure surface

represent points where minor and intermediate stresses swap places, i.e. �� , and blunt

corners are points where major and intermediate principal stresses become equal, i.e. �� .

Furthermore, it also merits noting that Eq. 4.6.4, in which the angle � is constrained such

that � �² ¶ � ¶ �

², gives only the section SB of the Hoek-Brown failure surface cross section in

Fig. 4.11 and other sections of this hexagonal cross section can be plotted considering the

symmetric properties of the failure criterion. The Hoek-Brown criterion is a periodic function

with the period of �� , and hence, its trace on the deviatoric plane is repeated every

�� radians.

Consequently, to plot, for example, the section BT in Fig. 4.11 requires replacing the angle �

in Eq. 4.4.9 with � [ ��, which results in the following relationship:

( )3

35

sin

sin3

sin

3

2 12

3

2

1IJ

+

+

+

+

=

πθ

πθ

πθ

σ

σ

σ

(4.6.8)

Substituting Eq. 4.6.8 into Eq. 4.6.1 results in formulating the Hoek-Brown criterion in terms

of invariants of the principal deviatoric stress tensor, similar to Eq. 4.6.4 with defining the

parameter ·/k in Eq. 4.6.4, as follows:

663

3sin2

)(

πθπθπ

λ ≤≤−

−=

i

HB

m

BTfor (4.6.9)

However, the parameterς remains unaltered. Other sections of the graph can be plotted in the

similar manner. Therefore, on the assumption of _� � DE&FCEF, the trace of a given failure

criterion can be plotted on a certain deviatoric plane. In order to reproduce the Hoek-Brown

cross sections on different deviatoric planes along the hydrostatic axis to produce the relevant

three-dimensional failure surface, as illustrated in Fig. 4.12, the parameter _� is assumed to be

a variable and is defined, considering Eq.4.6.4 for the Hoek-Brown criterion, as follows:


95

( )i

cHBcHBHBHB m

sJJI

222

13 σσλς −+

= (4.6.10)

The components � and _� of a failure stress point in the deviatoric stress space predicted by the

Hoek-Brown criterion are given by Eqs. 4.6.4 and 4.6.10 for various values of the angle � (�/k

and _�/k). In order to calculate the corresponding components of the failure stress point in the

principal stress space, the transformation matrix in Eq. 4.4.5 can be applied. After

transformation from the deviatoric stress space into the principal stress space of all failure

stress points predicted by the Hoek-Brown the three-dimensional failure surface associated

with the Hoek-Brown criterion can be plotted in the principal stress space (see Fig. 4.12). The

relevant MATLAB code for plotting the Hoek-Brown failure surface can be found in

Appendix F.

As is obvious from Eq. 4.6.1 the Hoek-Brown criterion incorporates only the minor principal

stress in rock failure stress (��) formulation and neglects the influence of the intermediate

principal stress on rock strength. In order to include the influence of the intermediate principal

stress to estimate the rock strength in three-dimensional stress state more precisely, a number

of three-dimensional failure criteria, based on the Hoek-Brown criterion have been introduced

over the past few decades. The reasons why the Hoek-Brown criterion has been adopted as a

σ2 (MPa)

σ1 (MPa)

σ 3 (M

Pa)

Figure 4.12 The Hoek-Brown criterion in the principal stress space


96

basis for developing new three-dimensional predictive models were outlined in Chapter 2,

Section 2.4.3.

4.6.2. The Pan-Hudson criterion

The strategy adopted by Pan and Hudson (1988), for developing a three-dimensional criterion

based on the Hoek-Brown criterion was to approximate the Hoek-Brown failure surface with a

conical surface the cross section of which on the deviatoric plane is a circle between the

inscribed and circumscribed circles to the hexagonal cross section of the Hoek-Brown

criterion (Fig. 4.13). According to Pan and Hudson (1988) for weak rock masses with small A

and & the hexagonal cross section of the Hoek-Brown on the deviatoric plane can be

approximated by a circle. Although this approximation produces a negligible error where

parameters A and & are small, errors cannot be ignored in the case of strong rocks for which

parameters A and & are relatively large. In other words, the Pan-Hudson criterion does not

work properly for good quality rock masses or intact rock. Furthermore, the Pan-Hudson

criterion does not reduce to the original form of the Hoek-Brown criterion where �� .

Consequently, as pointed out by Priest (2010), under triaxial compression (�� ' �� ) the

Pan-Hudson criterion does not predict the same value for the failure stress as the Hoek-Brown

criterion and other three-dimensional criteria developed based on the Hoek-Brown criterion.

Another drawback of the Pan-Hudson criterion is that the criterion does not calculate the

uniaxial strength of rock (��) under uniaxial compression (�� ' �� 0). The Pan-

Hudson criterion can be re-derived through the following procedure:

When �� , which corresponds to � � �² (Eq. 4.5.1), Eq. 4.6.2 reduces to:

033

23 212

2 =−

+

−−= ccibHB s

IJmJF σσ (4.6.11)

Where m/kB is a function which represents the points exactly located on the blunt corners of

the Hoek-Brown cross section on a certain deviatoric plane ( _� � DE&FCEF). Therefore, the

radius of the inscribed circle to the Hoek-Brown hexagonal cross section on the deviatoric


97

plane can be calculated by solving Eq. 4.6.11 for the term WV�, which is given also by Eq.

4.6.7. Furthermore, substituting � � � �² into Eq. 4.6.2 yields:

033

3 2122 =−

+

−−= ccisHB s

IJmJF σσ (4.6.12)

Where m/k� represents the points on sharp corners of the Hoek-Brown cross section on a

particular deviatoric plane and therefore, the radius of the circumscribed circle to the Hoek-

Brown criterion on the same deviatoric plane can be calculated by applying Eq. 4.6.12, as is

given also by Eq. 4.6.6. Accordingly, the function mc/ which represents a mean circle between

the inscribed and the circumscribed circles to the Hoek-Brown cross section can be expressed

as follows:

HBbHBbHBs

PH FFF

F +−=2

(4.6.13)

Expanding and rearranging Eq. 4.6.13, the equation of the mean circle or the Pan-Hudson

criterion, can be written as follows:

ciic

sI

mJmJ σσ

=−+32

33 122 (4.6.14)

Hoek-Brown Pan-Hudson

Deviatoric Plane

�� (MPa)

Circumscribed circle

�� (MPa)

Inscribed circle

Mean Circle (Pan-Hudson)

Figure 4.13 The cross section of the Hoek-Brown criterion on the deviatoric plane


98

The radius of the mean circle or the distance between the hydrostatic axis and the trace of the

Pan-Hudson criterion on the deviatoric plane is, therefore, given by:

+++−== s

ImJr

c

iPHPH

cPHPH σ

λλσ3

126

22 12

2 (4.6.15)

Where A6, s and �� are the Hoek-Brown parameters and the parameter ·c/ is given as √�� A6.

As illustrated in Fig. 4.13, the Pan-Hudson circle has six intersections with the Hoek-Brown

hexagonal cross section on the deviatoric plane. These intersections, however, do not coincide

with the apices of the Hoek-Brown hexagonal cross section where �� or �� . This

geometrical interpretation explains the reason why the Pan-Hudson criterion does not reduce

to the original form of the Hoek-Brown criterion when the stress state is assumed as two-

dimensional, i.e. �� or �� . Therefore, it is more appropriate to refer to the Pan-

Hudson criterion as a three-dimensional failure criterion which sources its input parameters

from the Hoek-Brown criterion, rather than a three-dimensional version of the Hoek-Brown.

The Pan-Hudson radius �c/ is given by Eq. 4.6.15 and therefore, the Pan-Hudson cross section

on the deviatoric plane can be plotted for _� � DE&FCEF. However, considering the parameter

_� as a variable the three-dimensional surface of the Pan-Hudson criterion can be plotted in the

principal stress space (Fig. 4.14).

σ2 (MPa)σ

1 (MPa)

σ 3 (M

Pa)

Figure 4.14 The Pan-Hudson criterion in the principal stress space


99

Considering Eq. 4.6.15 the parameter _� for the Pan-Hudson criterion can be defined as:

ci

cPHcPHPH m

sJJI

σσσ 2

221

39 −+= (4.6.16)

The relevant MATLB program, used for plotting the Pan-Hudson failure surface in the

principal stress space can be found in Appendix F.

4.6.3. The Zhang-Zhu criterion

As pointed out by Zhang and Zhu (2007), it is desirable for a three-dimensional version of the

Hoek-Brown criterion to be reduced to the original form of the Hoek-Brown criterion, when

�� or �� . In triaxial compression (�� ' �� ) and extension (�� ' ��)

stress states, the second invariant of the deviatoric tensor, V�, (Eqs. 4.2.7) reduces to:

( )3

231

2σσ −=J (4.6.17)

Substituting Eq. 4.6.17 into Eq. 4.6.14, the Pan-Hudson criterion reduces to the following

relationship:

( ) ( ) 231

231 2 cmci

ci smm σσσσσσσσ =−−+− (4.6.18)

The parameter �R in Eq. 4.6.18 is the mean normal stress GeI� M and is assumed to be constant.

Furthermore, rearranging Eq. 4.6.1, the Hoek-Brown criterion can also be written as:

( ) 23

231 cci sm σσσσσ =−− (4.6.19)

Substituting Eq. 4.6.19 into Eq. 4.6.18, the constant parameter �R can be defined so that Eq.

4.6.18 reduces to the original form of the Hoek-Brown criterion under triaxial compression

and extension states of stress. To satisfy this condition it is necessary to define the constant

parameter �R as follows:


100

231

2,σσσσ +== mm (4.6.20)

Consequently, if the parameter eI� in the Pan-Hudson criterion is replaced with the

parameter �R,� � HIJHL� , the resultant formulation predicts the same failure stress for intact

rock material as the Hoek-Brown criterion where �� and �� . Replacing the

parameter eI� in the Pan-Hudson criterion with �R,� yields a three dimensional version of the

Hoek-Brown criterion, which was first proposed by Zhang and Zhu (2007). Accordingly, the

Zhang-Zhu criterion for predicting the failure stress of intact rock in three-dimensional stress

state is expressed as:

cmiic

smJmJ σσσ

=−+ 2,22 2

33 (4.6.21)

Where the parameter �R,�, given by Eq. 4.6.20 can also be defined as follows:

2321

2,SI

m −=σ (4.6.22)

Furthermore, considering Eq. 4.4.11, the Eq. 4.6.22 can be written in the following form:

θσ sin3

3

321

2,JI

m −= (4.6.23)

Substituting Eq. 4.6.23 into Eq. 4.6.21, the Zhang-Zhu criterion can also be expressed as:

( ) 03

sin2332

3 1222

=

+−++= s

ImJmJF

c

i

c

i

cZZ σσ

θσ

(4.6.24)

Eq. 4.6.24 presents a more appropriate expression of the Zhang-Zhu criterion in the deviatoric

stress space. Considering the function m̧ ¸ in Eq. 4.6.24, the radial distance between the

hydrostatic axis and the trace of the Zhang-Zhu criterion on the deviatoric plane is calculated

as follows:

+++−== s

ImJr

c

iZZZZ

cZZZZ σ

λλσ

312

6

22 12

2 (4.6.25)


101

Where the parameter ·¸¸ for the Zhang-Zhu criterion is defined as:

( )θλ sin2332

+= iZZ

m (4.6.26)

The trace of the Zhang-Zhu failure surface on a deviatoric plane can be plotted by applying

Eqs. 4.6.25 and 4.6.26 and assuming the term _� as constant (Fig. 4.15).

In order to reproduce the Zhang-Zhu cross sections on different deviatoric planes along the

hydrostatic axis, to produce the associated three-dimensional failure surface, the Parameter _�

is assumed to be a variable and is defined, regarding to Eq. 4.6.25, for the Zhang-Zhu

criterion, as follows:

−+=

i

cZZcZZZZZZ m

sJJI

222

12

3σσλ

(4.6.27)

Considering Eqs. 4.6.25, 4.6.26 and 4.6.27 and by applying an iteration loop, the three-

dimensional failure surface corresponding to the Zhang-Zhu criterion can be plotted in the

principal stress space (see Fig. 4.16). The relevant MATLAB code for plotting the Zhang-Zhu

criterion can be found in Appendix F.

Zhang-Zhu Hoek-Brown

Deviatoric plane

�� (MPa)

�� (MPa)

Figure 4.15 The cross section of the Zhang-Zhu criterion on the deviatoric plane


102

4.6.4. Generalised Priest criterion

Combining the Drucker-Prager and the Hoek-Brown criteria Priest (2005) developed a three-

dimensional failure criterion. In the generalised Priest criterion the linkage between the

Drucker-Prager and the Hoek-Brown criteria was based on the assumption that both criteria

are supposed to give the uniaxial compressive strength of rock under uniaxial compression.

The first invariant of the principal stress tensor (_�) and the second invariant of the principal

deviatoric stress tensor (V�) reduce to the following expressions under uniaxial compression:

3

3

2

2

1

c

c

J

I

σ

σ

=

=

(4.6.28)

The Drucker-Prager criterion can be given as:

31

2BI

AJ += (4.6.29)

Substituting Eq. 4.6.28 into Eq. 4.6.29, parameters A and B can be given as follows:

σ2 (MPa)σ

1 (MPa)

σ 3 (M

Pa)

Figure 4.16 The Zhan-Zhu criterion in the principal stress space


103

( )( )

c

c

c

c

I

JB

I

JIA

σσ

σσ

−−

=

−−

=

1

2

1

21

33

3

33

(4.6.30)

Priest (2005) pointed out that for the two criteria to be compatible the following relationship

must be satisfied:

111 III DPHB == (4.6.31)

Where

32 31

1HBHB

HBIσσ += and

3321

1σσσ ++

= fDPI (4.6.32)

In Eq. 4.6.32 the terms ��0B and ��0B are the minor principal stress at failure and the failure

stress, calculated by means of the Hoek-Brown criterion through the following relationship:

a

c

HBcHBHB s

m

++=

σσσσσ 3

31 (4.6.33)

Furthermore, according to Priest (2005), the radial distance between the hydrostatic axis and

the cross section of the failure surface on the deviatoric plane must be identical for both the

Hoek-Brown and the Drucker-Prager criteria, that is:

rrr DPHB == (4.6.34)

Where

21

3

21

2

21

1

21

3

21

1

333

32

3

−+

−+

−=

−+

−=

IIIr

IIr

fDP

HBHBHB

σσσ

σσ

(4.6.35)

The parameter �� is the failure stress for the specified principal stresses �� and ��. Priest

(2005) adopted a numerical iteration to solve Eqs. 4.6.29 to 4.6.35 for seven unknowns,


104

namely ��/k , ��/k , A, B, _�, V� and ��. Melkoumian et al. (2009) developed an explicit

solution for the Generalised Priest criterion. Considering the closed form solution of the

generalised Priest criterion for intact rock material the major principal stress at failure is given

as:

( )3231 3 σσσσ +−+= PHB (4.6.36)

For intact rock parameter P is defined as:

2

1

3 1

+

=

c

HBic

mP

σσσ (4.6.37)

Where

( )F

FEEHB 22

232

232

3σσσσσ

−−−++=

m (4.6.38)

Parameters F and E are given as:

c

i

CE

CmF

σ21

21

2

3

=

+=−

(4.6.39)

Where

( )c

imC

σσσ

21 32 ++= (4.6.40)

Furthermore, under uniaxial compression the parameter V�/k, given by Eq. 4.6.4, is supposed

to be identical to V�, given by Eqs. 4.6.28. Substituting Eq. 4.6.28 into Eq. 4.6.4 gives the

following relationship:

131

33

sin2cos

34 2 +=

−+ i

im

m θπ

θ (4.6.41)


105

From Eq. 4.6.41 the angle � is calculated as �² . Substituting this value for � into Eq. 4.6.4 the

Priest criterion for intact rock can be expressed in terms of invariants of the deviatoric tensor

as follows:

ci

ic

sIm

JmJ σσ

=−+33

33 122 (4.6.42)

Furthermore, the radial distance from the hydrostatic axis to the trace of the generalised Priest

criterion on a deviatoric plane, given by _� � DE&FCEF, can be expressed as:

+++−== s

ImJr

c

iPP

cPGP σ

λλσ3

1262

2 122 (4.6.43)

The parameter ·c for the Priest criterion is defined as √�� A6. Considering Eq. 4.6.43 the cross

section of the generalised Priest failure surface can be plotted on the deviatoric plane (see Fig.

4.17).

In order to plot the corresponding three-dimensional failure surface of the Priest criterion (see

Fig. 4.18) through the same approach as that adopted for other three-dimensional criteria

(Appendix F), the Parameter _� is assumed to be a variable.

Hoek-Brown Priest criterion

Deviatoric plane

�� (MPa)

�� (MPa)

Figure 4.17 The cross section of the generalised Priest criterion on the deviatoric plane


106

From Eq. 4.6.43 the parameter _� for the generalised Priest criterion is defined as:

ci

cPciPGP m

sJmJI

σσσ 2

221

339 −+= (4.6.44)

4.6.5. The Simplified Priest Criterion

In addition to the generalised Priest criterion, in order to incorporate the influence of the

intermediate principal stress on the failure stress of rock, Priest (2005) proposed another three-

dimensional failure criterion based on the Hoek-Brown criterion. He introduced a weighting

factor w ranging from 0 to 1 and defined the minor principal stress (��/k) to be included in the

Hoek-Brown criterion as follows:

( )10

1 323

≤≤

−+=

w

wwHB σσσ (4.6.45)

As is inferred from Eq. 4.6.45, when w is 0 the intermediate principal stress (��) has no

influence and when w is 1 the minor principal stress (��) has no influence on rock strength.

Substituting Eq. 4.6.45 into the Hoek-Brown criterion gives the following formulation:

σ2 (MPa)σ

1 (MPa)

σ 3 (M

Pa)

Figure 4.18 The generalised priest criterion in the principal stress space


107

2

1

331

++= s

m

c

HBcHBHB σ

σσσσ (4.6.46)

Considering Eqs. 4.6.31 and 4.6.32, the simplified Priest criterion for evaluation of the rock

failure stress (��) in three-dimensional stress is given by the following formulation:

( )32311 2 σσσσσ +−+= HBHBf (4.6.47)

Substituting Eq. 4.4.9 into Eq. 4.6.47, the simplified Priest criterion can be written in terms of

invariants of the deviatoric stress tensor as follows:

03

122

2 =

+−+= s

ImJJF

c

iSP

cSP

cSP σ

λσ

ςσ

(4.6.48)

Where parameters -¹c and ·¹c are defined as follows:

( )

+−

+=

−++−

+=

6sin

3sin

3

32

6cos122sin3236

6sin36 222

πθπθλ

πθθπθς

wm

ww

iSP

SP

(4.6.49)

Considering Eqs. 4.6.48 and 4.6.49, the radial distance from the hydrostatic axis and the trace

of the simplified Priest criterion on the deviatoric plane can be expressed as:

+++−== s

ImJr

c

iSPSPSP

SP

cSPSP σ

ςλλς

σ3

42

22 12

2 (4.6.50)

Using Eqs. 4.6.49 and 4.6.50 the cross section of the simplified Priest criterion can be plotted

on the deviatoric plane. However, it should be noted that since the parameter w appears in

these equations, the shape of the cross section of the simplified Priest criterion changes with

the changes of the parameter w. Priest (2005) proposed that for a range of sedimentary and

metamorphic rocks the parameter w depends only on the minor principal stress (��) and could

be calculated from the following relationship:

15.0315.0 σ≈w (4.6.51)


108

Therefore, if the minor principal stress (��) takes the values of, for example, 10 MPa and 100

MPa, the weighting factor w is calculated as 0.211and 0.299, respectively. The cross sections

of the simplified Priest criterion on the deviatoric plane are given in Fig. 4.19 (a) and (b), for

cases when the least principal stress takes the values of 10 MPa and 100 MPa, respectively. As

is obvious from Fig. 4.19, under biaxial extension (�� ' ��), which is represented by

blunt corners in the Hoek-Brown cross section, the simplified Priest criterion does not

calculate the same value for the failure stress as the Hoek-Brown criterion. However, since the

Hoek-Brown criterion itself was developed as an empirical criterion based on a series of

conventional triaxial testes in which the stress condition is manifested by (�� ' �� ),

there is no evidence proving that the Hoek-Brown criterion accurately calculates the failure

stress under biaxial extension, i.e. when �� ' �� holds.

The three-dimensional failure surface of the simplified Priest can also be plotted for a given

value of �� and assuming the parameter _� as a variable, which, considering Eq. 4.6.50, can be

defined for the simplified Priest criterion as follows:

( )ci

cSPcSPSPSP

m

sJJI

σσσλς 2

221

3 −+= (4.6.52)

The relevant MATLAB code for plotting the three-dimensional failure surface, representing

the simplified Priest criterion in the principal stress space can be found in Appendix F. The

Figure 4.19 The cross section of the simplified Priest criterion on the deviatoric plane for (a) )� � � MPa, (b) )� � � MPa

Simplified Priest Hoek-Brown

(b)

Simplified Priest Hoek-Brown

(a)


109

effect of the weighting factor w (due to the change in the minor principal stress (��) values) on

the shape of the simplified Priest failure surface is illustrated in Fig. 4.20 (a) and (b).

4.7. Experimental Evaluations of Rock Behaviour under Three-

Dimensional Stress

Since the Hoek-Brown criterion is an empirical formulation which was developed and

formulated by Hoek and Brown (1980), based on comprehensive experimental studies on the

rock performance under conventional triaxial tests (�� ' �� ), it is also essential for

three-dimensional failure criteria, which have been derived based on the Hoek-Brown

criterion, to be validated against true-triaxial (�� ' �� ' ��) experimental data. Colmenares

and Zoback (2002) examined a number of selected failure criteria by comparing them with

published true-triaxial experimental data for five different rock types, namely Dunham

dolomite, Shirahama sandstone, Solnhofen limestone, Yuubari shale and KTB amphibolite.

They first defined a correlation coefficient to investigate the extent of dependency of the

failure stress of a particular rock type on the intermediate principal stress and then identified

that the Modified Wiebols and Cook (Zhou, 1994) and the Modified Lade (Ewy, 1999) criteria

are in reasonable agreement with experimental data, especially for those rocks with higher ��-

dependency of failure stress. However, it should be noted that some three-dimensional Hoek-

Brown based criteria which have been developed especially for rock material had not been

σ2 (MPa)σ

1 (MPa)

σ 3 (M

Pa)

(b)

σ2 (MPa)σ

1 (MPa)

σ 3 (M

Pa)

(a)

Figure 4.20 The Simplified Priest criterion in the principal stress space, for (a) )� � � MPa º � . �� and (b) )� � � MPa, º � �. ¬¬


110

introduced at that time and therefore, were not included in the comparative study by

Colmenares and Zoback (2002).

A statistical evaluation of the Hoek-Brown based three-dimensional failure criteria based on

nine sets of published true-triaxial test data was carried out in this study. Data sets selected

from true-triaxial experiments on nine different types of rocks, included three carbonates

(Solnhofen limestone, Dunham Dolomite and Yamaguchi marble) and four silicates (Mizuho

trachyte, Orikabe monzonite, Inada granite and Manazuru andesite) studied by Mogi (1971a),

also presented by Mogi (2007), Westerly granite by Haimson and Chang (2000) and KTB

amphibolite by Chang and Haimson (2000). All true-triaxial data were obtained from tests on

prismatic rock specimens. The dimensions of the rock specimens in Mogi’s experiments were

15×15×30 mm with the aspect ratio of 2, and for Westerly granite and KTB amphibolite the

specimen’s dimensions were 19×19×38 mm with the same aspect ratio of 2. Therefore, the

true-triaxial data in which the intermediate principal stress is zero (�� 0) can be applied for

determining the Hoek-Brown parameter A6 and Coulomb parameters ( and ?). Table 4.1

presents Coulomb input parameters ( and ?), uniaxial compressive strength (��) and the input

parameter A6 for the Hoe-Brown criterion for nine types of rocks discussed in this study.

True-triaxial data sets are presented in Appendix C, tables C.1 to C.9.

Rock specimen σc (MPa) c(MPa) φ (Deg) mi

Westerly Granite 165 34 52.6 38.6 KTB Amphibolite 201 31.3 48.5 37.3 Dunham Dolomite 261 64.2 37.6 9.7 Solnhofen Limestone 310 90.4 29.5 4.6 Yamaguchi Marble 82 20 37.8 10.3 Mizuho Trachyte 100 25.9 35.3 10.9 Manazuru Andesite 140 27.2 47.5 33.7 Inada Granite 229 46.4 46 29.5 Orikabe Monzonite 234 50.8 43 20

4.7.1. The influence of intermediate principal stress on failure stress

By plotting true-triaxial experimental data in �� domain, the ��-dependency of the

fracture strength can be qualitatively illustrated (Fig. 4.21). In order to quantify the extent to

Table 4.1 Hoek-Brown and Coulomb parameters of the rock types studied


111

which the fracture strength (��) depends upon the change in �� values for a constant ��

Colmenares and Zobak (2002) calculated the correlation coefficient between �� and �� using

linear Pearson’s correlation as:

[ ] [ ]21

2121

,,

σσ

σσσσSS

CovCorr = (4.7.1)

Where gHI and gHK are the standard deviation of �� and ��, respectively. Pearson’s correlation

lies between -1 and 1. When the Pearson correlation is 1, it indicates that �� increases linearly

with �� (perfect positive correlation) and when the Pearson correlation is -1, it means ��

decreases linearly as �� increases (anti-correlation). If the two variables are absolutely

independent, the Pearson correlation coefficient will be zero. However, the converse in not

always true, i.e. a zero Pearson’s correlation only postulates that there is no linear correlation

between the two variables, yet the variables may be correlated non-linearly. On the other hand,

as is obvious from Fig. 4.21, the relation between �� and �� is non-linear. Furthermore, if a

second order polynomial (quadratic) function is fitted to data points in the �� domain for

constant �� values, the coefficient of fitness to the quadratic function as a nonlinear correlation

coefficient is calculated as:

SST

SSEr −= 1 (4.7.2)

Where ‘SSE’ is the ‘Sum of the Squared Errors’ between the observed values of the fracture

stress (��"�B�$) and the fitted values of the fracture stress (��"6 $) given by:

( )2

1)(1)(1∑

=−=

N

iifitiobsSSE σσ (4.7.3)

The parameter ‘SST’ in Eq. 4.7.2 is known as the ‘Sum of the Squared Total’ and is calculated

as:

( )2

1)(1)(1∑

=−=

N

iiobsiobsSST σσ (4.7.4)


112

The parameter �»�"�B�$ is the mean value of ��"�B�$6. Therefore, since the experimental data do

not demonstrate a linear relationship between the fracture stress (��) and the intermediate

principal stress (��), calculation of the linear correlation coefficient between these two

variables does not provide any information about the dependence of �� on ��. However, if it is

assumed that the intermediate principal stress and the fracture stress are correlated through a

quadratic relationship, this correlation can be quantified by applying Eq. 4.7.2. Furthermore, in

order to visualise the extent of ��-dependency of the fracture stress at any given value for ��,

calculated values for the non-linear correlation coefficient can be plotted versus the minor

principal stress (Fig. 4.22).

The linear correlation coefficient between �� and �� calculated by the means of Pearson’s

linear correlation coefficient was also plotted versus the minor principal stress and is presented

in Appendix C, Fig. C.1. As illustrated in Fig. 4.22, correlation of fitness to quadratic

functions in almost all cases lies between 0.8-1, which indicates that the fracture stress closely

correlates with the intermediate principal stress through a quadratic function. This close

correlation suggests that the nature of dependency of the failure stress (��) on the intermediate

principal stress ( ��) can be well simulated by a quadratic function.

Furthermore, all quadratic functions fitted to data points are concaved downward, i.e. the

second derivatives of all quadratic functions are negative. Such quadratic functions grow to a

maximum as the variable increases, thereafter the function declines with the increasing

variable. Equating the first derivative of each one of these quadratic functions for a given

value for �� to zero, the value of the intermediate principal stress after which increasing ��

negatively affects the rock strength can be calculated. The application of the linear correlation

coefficient can only be meaningful if two correlation functions are calculated; the first one

from the point �� to the point �� at which the quadratic function is at its maximum,

and the second one from the point �� down to the failure point, which is higher than the

point �� (Fig. 4.21). It merits noting that the former yields a positive correlation and the

latter gives an anti-correlation.

113

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800

σ1

(MP

a)

KTB amphiboliteσ₃=0 σ₃=30 σ₃=60 σ₃=100 σ₃=150

0 100 200 300 400

Westerly graniteσ3=0 σ3=2 σ3=20 σ3=38

σ3=60 σ3=77 σ3=100

0 200 400 600σ2 (MPa)

Inada graniteσ3=20 σ3=40 σ3=100 σ3=150 σ3=200

0

200

400

600

800

1000

1200

1400

0 200 400 600

σ1

(MP

a)

Dunham dolomiteσ3=25 σ3=45 σ3=65σ3=85 σ3=105 σ3=125σ3=145

0 100 200 300 400

Manazuru andesiteσ3=20 σ3=40 σ3=70

0 200 400 600σ2 (MPa)

Orikabe Monzoniteσ3=40 σ3=80 σ3=140 σ3=200

Figure 4.21 Fitting quadratic functions to true-triaxial experimental data in )� � )� domain (continues)

114

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250

σ1

(MP

a)

Yamaguchi marbleσ3=12.5 σ3=25 σ3=40

0 200 400 600

Solnhofen limestoneσ3=20 σ3=40 σ3=60 σ3=80

0 200 400 600σ2 (MPa)

Mizuho trachyte

σ3=45 σ3=60 σ3=75 σ3=100

Continued from Figure 4.21 Fitting quadratic functions to true-triaxial experimental data in �� domain


115

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140 160

Non

-line

ar C

orre

latio

n C

oeff

icie

nt b

etw

een σ

1an

d σ

2

σ3 (MPa)

KTB Amphibolite

Westerly Granite

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140

Non

-line

ar C

orre

latio

n C

oeff

icie

nt

betw

een σ

1an

d σ

2

σ3 (MPa)

Dunham Dolomite

Solenhofen Limestone

Yamaguchi Maeble

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

Non

-line

ar C

orre

latio

n C

oeffi

cien

t bet

wee

n σ

1an

d σ

2

σ3 (MPa)

Mizuhu TrachyteManazuru AndesiteInada GraniteOrikabe Monzonite

Figure 4.22 Non-linear correlation coefficient between the failure stress ()�) and the intermediate principal stress ()�) versus the least principal stress ()�)


116

4.7.2. A modification to the simplified Priest criterion

If the failure stress in the principal stress space is represented by a point P, as illustrated in

Fig. 4.6, the length OC along the hydrostatic axis from the origin of the principal stress space

perpendicular to a deviatoric plan containing the point P is calculated as:

( )3213

1 σσσ ++=POC (4.7.5)

Furthermore, the radial distance from the hydrostatic axis to the point P on the deviatoric

plane can be given as follows:

( ) ( ) ( )

−+−+−== 2

132

322

212 3

12 σσσσσσJrP (4.7.6)

On the other hand, the Hoek-Brown criterion predicts a failure stress, which can also be

represented by a point such as HB in the principal stress space. For such a stress point the

length OCHB along the hydrostatic axis and the radial distance �/k from the hydrostatic axis

can be given by the following expressions, recalling that the Hoek-Brown criterion assumes

equal values for the intermediate and the minor principal stresses.

( )

( ) )(3

22

)(23

1

312

31

bJr

aOC

HBHBHBHB

HBHBHB

σσ

σσ

−==

+=

(4.7.7)

In Eqs. 4.4.7, ��/k and ��/k are calculated from Eqs. 4.6.45 and 4.6.46, respectively. In order

to develop a three-dimensional failure criterion based on the Hoek-Brown criterion, it is

desired that the failure point HB, predicted by the Hoek-Brown criterion, coincides with the

failure point P which occurs under the three-dimensional stress condition. Therefore, the stress

vector � c in Fig. 4.6, which indicates the position of the failure stress in the principal stress

space, must be identical with the stress vector � /k calculated by the means of the Hoek-Brown

criterion:

HBP σσ rr = (4.7.8)


117

Therefore, considering Fig. 4.6 and recalling the Pythagoras theorem the Eq. 4.7.8 can be

written as:

2222HBHBPP OCrOCr +=+ (4.7.9)

Rearranging Eq. 4.7.9, the failure stress �� can be calculated as follows:

2

123

22

23

211 2

+−+= σσσσσ HBHBf (4.7.10)

It should be noted that the prediction accuracy of the failure criterion given by Eq. 4.7.10

depends upon the weighting factor w which appears in the formulation of ��/k (Eq. 4.6.45)

and consequently, in the formulation of ��/k. One strategy for formulating the weighting

factor w is to substitute Eq. 4.6.45 into Eq. 4.6.38, which results in the following relationship:

( )( )32

232

2

221

σσσσ

−−−−

+=F

FEEw

m (4.7.11)

Where parameters E and F are given by Eqs. 4.6.39. In a given stress state where �� and ��

are known by substituting the weighting factor w from Eq. 4.7.11 into Eq. 4.7.10 the failure

stress is calculated the same way as is done by the generalised Priest criterion. However, it

merits noting that the generalised Priest criterion assumes, as does the Drucker-Prager

criterion, a more profound strengthening effect for the intermediate principal stress than that

which is observed in true-triaxial experiments and therefore, tends to overestimate the rock

strength, especially at higher values of ��.

This overestimating tendency of the generalised Priest criterion can be addressed considering

the circular cross section of this criterion on the deviatoric plane. This circular cross section

indicates that the radial distance between the hydrostatic axis and the predicted failure point on

the trace of the failure surface on the deviatoric plane is identical for all stress conditions.

However, from Fig. 4.22 it can be observed that the failure stress decreases after the

intermediate principal stress grows closer to the major principal stress. Therefore, the radial

distance between the hydrostatic axis and the trace of the failure surface on the deviatoric


118

plane is supposed to be smaller when �� , i.e. at obtuse corners in the hexagonal cross

section of the Hoek-Brown criterion on the deviatoric plane.

Accordingly, in order to incorporate the dual nature of ��-dependency of the failure stress, the

weighting factor w can be determined so that the strengthening effect of the intermediate

principal stress to be addressed in accordance with the true-triaxial test data. It should be noted

that on a particular deviatoric plane the radial distance between the failure point P and the

failure point predicted by the Hoek-Brown criterion are supposed to be identical, so according

to Eqs. 4.7.6 and 4.7.7 (b) the following relationship holds:

( ) ( ) ( ) ( ) 231

213

232

221 2 HBHB σσσσσσσσ −=−+−+−

(4.7.12)

On the other hand, considering Fig. 4.6, for a particular deviatoric plane the distance OCP for

the failure point P and the distance OCHB for the point HB predicted by the Hoek-Brown

criterion are equal and therefore, recalling Eqs. 4.7.5 and 4.7.7 (a) the major principal stress

can be given as follows:

32311 2 σσσσσ −−+= HBHB (4.7.13)

Substituting Eq. 4.7.13 into Eq. 4.7.12, and after series of expansion and rearrangement, the

following relationship can be derived for calculating the weighting factor w for intact rock:

( ) ( ) ( ) ( ) 0145648154189 234 =−++−+−−+++− µηµηµηη wwww (4.7.13)

Where parameters � and � are defined as follows:

( ) ( ) 32232

22

32 and σσ

σσσσσµ

σσση ≠

−+=

−= ccc mm (4.7.14)

However, substitution of the weighting factor w, given by Eqs. 4.7.13 and 4.7.14, into Eq.

4.7.10 again results in overestimation of the failure stress and the predicted values for the

failure stress are quite similar to the predicted failure stress calculated by the generalised Priest

criterion. It is also important to note that when �� Eq. 4.6.45 reduces to ��/k � �� and

consequently the rock failure stress, �� in Eq. 4.7.10 will be the same as the failure stress

predicted by the original two-dimensional formulation of the Hoek-Brown criterion. An


119

alternative approach for determining the weighting factor w is to define this parameter so that

to achieve minimum misfit with true-triaxial data. In order to achieve this goal, the parameter

w was first defined to obtain exact match with experimental data for each case. On the other

hand, the parameter w can be assumed to be proportional to parameters � and �, (which appear

in Eq. 4.7.13 and are given by Eqs. 4.7.14), and to the minor principal stress (��) and the

uniaxial strength (��) as follows:

−∝

cw

σσ

ηµ 3 (4.7.15)

The relationship between the weighting factor w and parameters �, �, �� and �� can be

formulated as a power function by plotting the values of w for which Eq. 4.7.10 predicts the

exact value of the failure stress versus the values of the term G¼½ � HL

H¾M for each stress state

(Fig. 4.23) as follows:

5.0324.0

−=

cw

σσ

ηµ

(4.7.16)

Substituting the parameter w, as is given by Eq. 4.7.16, into the proposed three-dimensional

failure criterion given by Eq. 4.7.10, the dual nature of the ��-dependency of the failure stress

is appropriately taken into account. A statistical comparison of three-dimensional failure

criteria which source their input parameters from the Hoek-Brown criterion was carried out

against nine sets of published true-triaxial experimental data and is presented in the

subsequent section.

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

w

(µ/�)-(σ3/σc)

Figure 4.23 Actual values of the weighting factor º versus values of the term ¿À � )�

)P


120

4.7.3. Comparison of three-dimensional, Hoek-Brown based failure criteria

The common method used to compare the prediction accuracy of three-dimensional failure

criteria to experimental data is to create a two-dimensional graph plotting �� versus �� for a

constant �� (Appendix D, �� vs. �� graphs). A qualitative analysis is then undertaken to

assess how well the various criteria fit the experimental data in ��-�� domain. The key

limitation of this method is that each analysis only takes into account a single slice of the

three-dimensional stress space for a constant ��.

A more comprehensive error analysis can be conducted by giving consideration to three-

dimensional failure surfaces of failure criteria in principal stress space. Any point on the

failure surface represents a failure stress, which has been predicted by the associated failure

function and can be addressed by a stress vector (� ). The stress vector form the origin of the

principal stress space to the predicted stress point on the failure surface is depicted as � "R��Á�$

in Fig. 4.24. Similarly, an observed data point in the principal stress space can be represented

by a stress vector as � "�B�Á�,Á�$, as illustrated in Fig. 4.24.

The magnitude of the resulting vector, � "R��Á�$ � � "�B�Á�,Á�$, quantifies the difference

between the predicted and observed failure stresses and therefore, can be interpreted as a

measure of prediction accuracy of the failure model. Furthermore, negative values of the

subtraction resultant vector indicate that the relevant model underestimates the rock failure

stress and positive values of this vector are interpreted as that the rock strength has been

overestimated. Therefore, the prediction accuracy (PA) of a failure criterion, as a predictive

model, can be defined as follows:

)()( observedmodelPA σσ rr −= (4.7.5)

However, as the values of �� and �� are identical for both � "R��Á�$ and � "�B�Á�,Á�$, the only

distance relating to an error is in the �� direction. Therefore, Eq. 4.7.5 can be reduced to the

following expression:

)(1)(1 observedmodelPA σσ −= (4.7.6)


121

In addition, absolute percentage error can be defined as:

100)(1

)(1)(1 ×−

=observed

observedmodelE

σσσ

(4.7.7)

Comparing the calculated values of absolute percentage errors from Eq. 4.7.7 for each one of

three-dimensional criteria incorporated in this study it was revealed that in all cases (with the

exception for Yamaguchi Marble) the proposed criterion, given by Eq. 4.7.10, namely the

modified simplified Priest, predicts the rock failure stress more accurately than other three-

dimensional models. Results of this statistical comparison are presented in Table 4.2. After the

proposed criterion the Zhang-Zhu criterion is in good agreement with the experimental data,

and in the case of Yamaguchi marble is more accurate than other models, including the

modified simplified Priest criterion. Furthermore, the relative likelihood of being zero for PA,

which is a measure of difference between the experimental and predicted failure stress values

and is given by Eq. 4.7.6, can be described using the probability density function (PDF) which

is parameterised in terms of the ‘mean’ and ‘variance’ as follows:

( )2

2

22

2

1, σ

µ

πσσµ

−−=

PA

ePAf (4.7.8)

)�� "ÂOÃÄÅ$

)�� "OÆÇÄ�ÈÄÃ$

��

��

��

)�� "ÂOÃÄÅ$ � )�� "OÆÇÄ�ÈÄÃ$

Deviatoric plane

Figure 4.24 Difference between predicted and observed failure stresses


122

Where � and � are the mean and variance, respectively. The probability density functions,

given by Eq. 4.7.8, were plotted versus PA for the nine data sets and six selected three-

dimensional failure criteria which are presented in Appendix D. The values for mean and

variance were also included in Figs. D.1to D.9, in Appendix D.

Manazuru Andesite

Criterion Rank σ₁(model)-σ₁(observed) (MPa) over/underestimation (average)

Average percentage error

(%) Modified Simplified Priest 1 4.15 overestimation 5.01

Simplified Priest 2 -16.11 underestimation 6.09 Zhang-Zhu 3 16.80 overestimation 6.79

Hoek-Brown 4 -100.04 underestimation 15.76 Generalised Priest 5 106.24 overestimation 17.83

Pan-Hudson 6 -170.01 underestimation 34.75

Inada Granite



(%) Modified Simplified Priest 1 -11.58 underestimation 3.73

Simplified Priest 2 -31.17 underestimation 4.34 Zhang-Zhu 3 -12.92 underestimation 4.67

Generalised Priest 5 59.83 overestimation 8.34 Hoek-Brown 4 -119.57 underestimation 12.99 Pan-Hudson 6 -336.48 underestimation 34.94

Orikabe Monzonite




Zhang-Zhu 3 -25.53 underestimation 6.41 Generalised Priest 5 45.28 overestimation 7.91 Simplified Priest 2 -54.12 underestimation 8.55

Hoek-Brown 4 -47.53 underestimation 9.76 Pan-Hudson 6 -251.34 underestimation 31.80

Yamaguchi Marble



(%) Zhang-Zhu 1 1.59 overestimation 3.88

Modified Simplified Priest 2 -3.33 underestimation 4.35 Simplified Priest 3 -23.35 underestimation 10.43


Continues

Table 4.2 Comparison of 3D Hoek-Brown based criteria


123

KTB Amphibolite




Zhang-Zhu 2 64.73 overestimation 13.65 Simplified Priest 3 -15.70 underestimation 15.91


Generalised Priest 6 216.71 overestimation 34.73

Westerly Granite





Hoek-Brown 4 -109.73 underestimation 17.97 Generalised Priest 5 112.70 overestimation 23.80


Dunham Dolomite




Zhang-Zhu 2 -7.43 underestimation 4.29 Simplified Priest 3 -45.30 underestimation 7.02


Solnhofen Limestone






Mizuho trachyte








124

4.8. True-triaxial Experiments at the University of Adelaide

Although a number of true-triaxial experiments have been conducted over the past decades, as

a major limitation in studying the rock performance under three-dimensional stress can still be

mentioned the lack of adequate true-triaxial experimental data in order to validate the

theoretical and empirical rock failure models. The main reasons for such limitation are the

elaborate rock specimen preparation techniques and complicated testing procedures.

According to Mogi (1971a) the major difficulty of the true-triaxial testing is ensuring the

application of three orthogonal, independent yet homogeneous stresses to rock specimens.

Nevertheless, there exists a serious need for further true-triaxial experiments to be conducted

on rock material. Considering this serious demand a number of polyaxial tests were conducted

at the University of Adelaide in collaboration with a group of four honours students. The main

purposes of conducting true-triaxial experiments were to validate three-dimensional rock

failure criteria developed based on the Hoek-Brown criterion, and to investigate the effect of

the specimen size and shape on the apparent strength of the rock specimen.

4.8.1. Experimental setup

True-triaxial apparatus

The true-triaxial cell used for testing was designed and fabricated at the University of

Adelaide by Prof. Stephen D. Priest and Dr. Nouné S. Melkoumian in collaboration with Mr

Adam Schwartzkopff et al. (2010) (Fig. 4.25). The design of the true-triaxial cell was adopted

from and is similar to an existing design outlined by King et al. (1997) (Schwartzkopff et al.,

2010). Lateral confining pressures are applied independently and orthogonally by two sets of

hydraulic jacks mounted on a steel reaction ring by means of intermediate jack support units.

Each opposing set of jacks is controlled by a hydraulic circuit, which is pressurised by a hand

pump connected to an adjustable pressure relief valve. Each jack has a capacity of 718 KN,

which corresponds to maximum pressures of 287, 200, 147 and 112 MPa on 50, 60, 70 and 80

mm sized cubic specimens, respectively. The hydraulic jacks control the extension of circular

pistons on which intermediate platen base units are mounted using rare earth magnets. The

platen base unit has an indention into which the platen is inserted.


125

One important feature of the true-triaxial cell is that replaceable platens of various sizes render

the cell capable of testing cubic specimens of different sizes, with dimensions of 50, 60, 70

and 80 mm. All different size platens were designed with the same dimensions at their base, to

allow them to be slotted into the platen base unit and the platen dimensions change from the

base to the contact surface to match the surface dimensions of the specimen. In order to avoid

the grinding of the platens which could result in damaging the cell, the platens were designed

with a 2% bevel around the edges to account for the deformation of the rock specimen during

testing (Schwartzkopff et al., 2010).

Top and bottom platens were also fabricated for each specimen size and the contacting

surfaces were designed to have the same chamfering as the lateral platens. Axial load is

applied using a compression machine which has a maximum capacity of 5000 KN. Strains in

the three principal stress directions are also monitored during testing by means of Linear

Variable Differential Transducers (LVDTs). The loading system was thoroughly calibrated

using a strain-gaged aluminium sample of known elastic properties. A detailed description of

the design, fabrication and calibration procedures of the true-triaxial cell at the University of

Adelaide is given in Schwartzkopff et al. (2010).

Figure 4.25 True-triaxial apparatus of the University of Adelaide [after Schwartzkopff et al.(2010)]


126

Specimen preparation

Specimens were made from Kanmantoo bluestone, which is a hard fine-grained, medium to

dark grey-blue meta-siltstone of early Cambrian age. The stone was sourced from the

Kanmantoo Stone Quarry in the Adelaide Hills. The bluestone occurs within the Tapanappa

formation within the larger Kanmantoo group within the Adelaide fold belt geological

providence. Kanmantoo bluestone was chosen because it is a relatively homogeneous rock.

This homogeneity reduces the risk of inaccurate test results due to flaws in individual

specimens.

To determine the uniaxial compressive strength, ��, and the parameter A6 for the Hoek-Brown

criterion a number of cylindrical specimens with aspect ratio of 2.4 were prepared to be tested

in the Hoek cell. Cores were drilled out from a block of Kanmantoo bluestone and were cut to

make specimens of 100mm in length and 42mm in diameter. The top and bottom faces of the

cores then were ground to minimise the parallelism offset.

To prepare cubic specimens of 60×60×60 mm dimensions to be tested in the true-triaxial

apparatus, first, cores of 85mm diameter were drilled out of the Kanmantoo block. Once the

cores were prepared, they were cut into rectangular prisms and then two cubes were cut out of

each prism (Fig. 4.26). Cubic specimens (60×60×60 mm) were surface ground to obtain

dimensions within 1mm from the prescribed size, with 0.05 mm of parallelism and

orthogonality offsets. It was a requirement that the cubic specimens were slightly larger than

their specified size to allow the cubes to be used in the true-triaxial cell without any contact

between platens in the true-triaxial cell. Undersized or exact sized cubes may cause contact

between platens.

Figure 4.26 Block of Kanmantoo Blue stone and preparation of cubic specimens [after Dong et al., (2011)]


127

True-triaxial tests

A total number of 10 true-triaxial tests were carried out on 60×60×60 mm cubic specimens of

Kanmantoo blue stone. True-triaxial tests were conducted at the University of Adelaide and in

collaboration with a group of four honour students. True-triaxial testing procedure consisted of

simultaneously raising all three principal stresses at a constant rate until �� reached its

prescribed value. Thereafter, the other two principal stresses (��and ��) were increased at the

same rate until �� reached its predetermined magnitude. From this point �� and �� were kept

constant and �� alone was raised until the specimen failed. Unloading was carried out after ��

decreased approximately 5%-10% of its peak level. The results of true-triaxial tests are

presented in Table 4.3.

Test No. σ₁ (MPa) σ₂ (MPa) σ₃ (MPa) I1 (MPa) √J2 (MPa)

1 450.62 21.07 21.43 493.12 247.90 2 426.92 49.16 19.80 495.87 227.05 3 585.70 70.61 22.63 678.93 312.16 4 537.80 41.54 42.65 621.99 286.19 5 582.93 74.73 42.94 700.59 303.00 6 673.06 101.39 40.95 815.40 348.81 7 771.82 59.68 62.66 894.16 410.30 8 792.79 90.75 63.19 946.72 413.51 9 701.27 118.27 65.02 884.56 352.97 10 761.87 140.04 63.20 965.11 383.13

Different types of failure were observed from the true-triaxial tests. The most common failure

mode from true-triaxial testing was a V-shaped crack (Fig. 4.27 (a)). This failure mode

involved the formation of two distinct cracks running through the cubic specimen. Another

failure mode observed from true-triaxial testing was an M-shaped crack (Fig. 4.27 (b)). It is

important to mention that regardless of the mode of failure, failure planes were in the direction

of the intermediate principal stress, ��, as was expected, and the specimens were separated out

in the direction of the minimum principal stress, ��.

Table 4.3 True-triaxial experimental data of Kanmantoo Bluestone, The University of Adelaide (2011)


128

Comparison and validation of three-dimensional failure criteria against true-triaxial data

Six rock failure criteria were selected and studied in this thesis. These criteria have been

developed based on the Hoek-Brown failure criterion in the sense that their input parameters

are the same as those for the Hoek-Brown criterion, namely the uniaxial compressive strength

and the Hoek-Brown constant parameter m. In order to evaluate and validate the selected

three-dimensional rock failure criteria against the true-triaxial experimental data, the uniaxial

compressive strength and the Hoek-Brown constant parameter m for Kanmantoo bluestone

had to be determined. Uniaxial tests were performed on cylindrical and cubic specimens of

Kanmantoo bluestone. The aspect ratio (length/diameter) for cylindrical specimen was 2.4 and

for the cubic specimen was 1 (Table 4.4).

Rock specimen Aspect Ratio (Length/Diameter)

UCS (MPa)

Average UCS (MPa)

cylinder 2.4 148.858 147.11343 cylinder 2.4 149.255

cylinder 2.4 143.228 cubic (60 × 60 mm) 1 190.278 190.27778 cubic (50 × 50 mm) 1 208.773 208.7727

Figure 4.27 (a) The V-shaped failure mode and (b) the M-shaped failure mode [after Dong et al. (2011)]

(a) (b)

Table 4.4 Uniaxial compressive strength of cylindrical and cubic specimens of Kanmantoo bluestone.


129

It is important to remember that apparent strength in short specimens becomes higher due to

the clamping effect at the two ends of the specimen. The underlying reason for this

phenomenon is that the loading machine does not deform and expand as much as the rock does

under the loading. The expansion of rock at machine-specimen interface exerts an additional

confining pressure on the rock specimen. This confining pressure is the result of the frictional

force acting on the interfacial area between the rock specimen and the loading machine and

restricts the lateral expansion of the rock specimen at the two ends of the specimen. It is

evident that with the increase of the length/diameter ratio, this effect should decrease gradually

and disappear at some critical value. Above this critical value, the strength should remain

constant and should represent the true strength under uniform compression. According to

Mogi (2007), this critical value for length/diameter ratio is about 2.5. Therefore, considering

Table 4.4, the uniaxial compressive strength of the Kanmantoo bluestone is determined as 147

MPa.

Conventional triaxial tests were also conducted on similar core samples, using the Hoek cell,

to determine the empirical Hoek-Brown parameter m for the Kanmantoo bluestone. Confining

pressure was applied at the same rate as the vertical load. Once the desired confining pressure

was reached it was maintained constant by bleeding off excess pressure as necessary. Since

the specimen tends to expand laterally as it is loaded vertically, confining pressure can become

too high if it is not bled off. Vertical loading continued at a constant rate until the specimen

failed. Hoek cell tests were performed under 5, 10 and 15 MPa confining pressures, as it is

given in Table 4.5.

Confining Pressure (MPa)

Failure Stress σ1 (MPa) σ3/σc [(σ1-σ3)/σc]

2

5 212.556 0.034 1.994 10 232.051 0.068 2.282 15 241.533 0.102 2.375

The empirical parameter m can be determined by rearranging the Hoek-Brown criterion (Eq.

4.6.1) as follows:

Table 4.5 Conventional triaxial tests for determining the Hoek-Brown constant parameter m


130

132

31 +

=

−

ccm

σσ

σσσ

(4.8.1)

By plotting 8"�� $/��:� against "��/��$, for the experimental data (Table 4.5), the

parameter m is determined as the slope of a best fit line, which intersects with the 8"�� $/��:�-axis at 1, as illustrated in Fig. 4.28. Therefore, considering the Hoek cell tests the

empirical parameter m is determined as equal to 16.131. However, the coefficient of

determination, 9� value of -2.86 indicates that the line is not well fitting the data.

Inserting the uniaxial compressive strength, ��, as 147 MPa and the Hoek-Brown constant m

as 16.131 into the selected three-dimensional rock failure criteria, the rock failure stress can be

calculated by means of each failure criterion (Table 4.6). However, using these values for ��

and m it was found out that all three-dimensional failure criteria underestimate the strength of

the cubic rock specimen under three-dimensional stress, as illustrated in ��-�� plots in Fig.

4.29.

y = 16.131x + 1

R² = -2.867

0.000

0.500

1.000

1.500

2.000

2.500

3.000

0.000 0.020 0.040 0.060 0.080 0.100 0.120

[(σ

1-σ

3)/σ

c]2

σ3/σc

Figure 4.28 Best fit line to conventional triaxial data for determining the Hoek-Brown constant parameter m

131

σ2 σ3 σ1

True-triaxial data

σ1 Hoek-Brown

σ1 Pan-Hudson

σ1 Zhang-Zhu

σ1 Generalised

Priest

σ1 Simplified

Priest

σ1 Modified Simplified

Priest 0.00 0.00 147.00 147.00 48.64 147.00 147.00 147.00 147.00

21.07 21.43 450.62 290.64 156.87 290.06 289.66 290.37 291.45 49.16 19.80 426.92 281.73 210.52 324.41 353.85 302.60 333.63 70.61 22.63 585.70 297.08 254.76 360.98 405.29 329.43 369.71 41.54 42.65 537.80 393.14 245.11 391.65 390.71 392.38 395.45 74.73 42.94 582.93 394.41 305.79 433.59 458.52 415.01 448.16 101.39 40.95 673.06 385.64 343.49 455.52 501.63 423.40 466.79 59.68 62.66 771.82 475.38 317.15 471.73 469.58 473.47 479.48 90.75 63.19 792.79 477.43 370.80 508.94 527.78 494.47 528.73 118.27 65.02 701.27 484.47 415.62 541.77 576.70 516.38 558.67 140.04 63.20 761.87 477.47 442.21 556.64 606.53 522.54 570.16

0

100

200

300

400

500

600

700

800

900

0 20 40 60 80

σ1

(MPa)

σ2 (MPa)

Kanmantoo bluestone, σ3 = 20 MPa

30 50 70 90 110

σ2 (MPa)


40 60 80 100 120 140 160

σ2 (MPa)

Kanmantoo bluestone, σ3 = 60 (MPa)

data Point

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

MSP

Table 4.6 Predicted values of failure stress by the means of each selected failure criteria for m = 16.131 and )P= 147 MPa

Figure 4.29 )�-)� plots, demonstrating that all 3D failure criteria underestimate the strength of the rock specimen


132

As is obvious form Table 4.6 and Fig. 4.29 the measured failure stress of cubic specimens of

Kanmantoo bluestone is about 30% (in the case of the Pan-Hudson criterion 50%) higher than

the predicted values of the failure stress of bluestone. One of the reasons for such

discrepancies can be the clamping end effects caused by friction on the steel-rock interface. In

order to eliminate the clamping end effect caused by friction on the steel-rock interface, thin

layers of HDPE plastic were applied between the rock end surface and the steel platens. In this

case, the failure mode can be described as two vertical cracks developed starting from the end

surfaces of the rock sample and in the direction of the intermediate principal stress. The cubic

specimen was separated out in the direction of the minimum principal stress and the failure

stress was significantly low. This phenomenon seems to occur because of the intrusion of the

plastic layer into the rock specimen (Fig. 4.30) and therefore this method is not recommended.

Another hypothesis can be described as the effect of the size and shape of the cubic rock

specimens on the apparent strength of the rock, measured using the true-triaxial cell. The

effect of the specimen’s size and shape can be acknowledged by adjusting the input

parameters of the three-dimensional failure criterion. As explained previously, the uniaxial

compressive strength of the Kanmantoo bluestone measured by conducting uniaxial

compression test on the 60×60×60 mm cubic specimen cannot be relied upon due to the

pronounced clamping end effects. Therefore, it is recommended that the uniaxial compressive

strength, ��, of the Kanmantoo blue stone be determined from the uniaxial compression tests

on cylindrical specimens with aspect ratio of 2.4. On the other hand, the empirical parameter

m can be determined for the cubic specimens of Kanmantoo bluestone tested in the true-

Figure 4.30 Intrusion of the HDPE plastic layer into the rock specimen [after Dong et al (2011)]

Marks due to the intrusion into the

specimen


133

triaxial cell. For this purpose, true-triaxial tests in which �� o �� can be adopted and the

corresponding �� and �� values can be substituted into Eq. 4.81, as presented in Table 4.7.

Confining Pressure (MPa)

Failure Stress σ1 (MPa) σ3/σc [(σ1-σ3)/σc]

2

21.07 450.62 0.11 5.10 41.54 537.80 0.22 6.80 59.68 771.82 0.31 14.00

It is also important to note that since the empirical parameter m is determined to account for

the size and shape effects of the cubic specimens on the apparent rock strength measured in

true-triaxial tests, the apparent uniaxial compressive strength of the 60×60×60 mm cube is

substituted into Eq. 4.8.1 for determining the parameter m. It merits emphasising that the

uniaxial compressive strength of the 60×60×60 mm cube must not be used as an input

parameter for three-dimensional criteria, but only for determining the empirical parameter m.

Plotting 8"�� $/��:� against "��/��$, for the experimental data (Table 4.7), the parameter

m is determined as the slope of the best fit line, which intersects with the 8"�� $/��:�-axis

at 1, as illustrated in Fig. 4.31.

Inserting the uniaxial compressive strength, ��, as 147 MPa and the Hoek-Brown constant m

as 36.6 into the selected three-dimensional rock failure criteria, the rock failure stress can be

calculated by means of each failure criterion, as presented in Table 4.8 and illustrated in Fig.

4.32.

y = 36.632x + 1

R² = 0.8408

0

2

4

6

8

10

12

14

16

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

[(σ

1-σ

3)/σ

c]2

σ3/σc

Table 4.7 Triaxial test data on cubic rock specimens for determination the empirical parameter m

Figure 4.31 Best fit line to triaxial test data on cubic specimens for determining the empirical parameter m

134

σ2 σ3 σ1

True-triaxial data

σ1 Hoek-Brown

σ1 Pan-Hudson

σ1 Zhang-Zhu

σ1 Generalised

Priest

σ1 Simplified

Priest

σ1 Modified Simplified

Priest/MSP 0.00 0.00 147.00 147.00 23.48 147.00 147.00 147.00 147.00

21.43 21.07 450.62 388.42 153.37 389.27 389.93 290.37 392.05 49.16 19.80 426.92 377.76 220.82 440.48 487.45 302.60 451.31 70.61 22.63 585.70 401.26 277.80 494.95 564.04 329.43 503.87 42.65 41.54 537.80 536.64 263.90 538.69 540.19 392.38 545.39 74.73 42.94 582.93 545.57 341.79 599.90 638.68 415.01 617.22 101.39 40.95 673.06 532.81 391.83 631.46 701.96 423.40 642.56 62.66 59.68 771.82 645.11 355.59 649.99 653.38 473.47 664.54 90.75 63.19 792.79 664.51 424.83 706.94 736.03 494.47 730.19 118.27 65.02 701.27 674.47 483.70 752.40 805.70 516.38 770.48 140.04 63.20 761.87 664.56 519.74 773.72 849.02 522.54 785.58

0

100

200

300

400

500

600

700

800

900

0 20 40 60 80

σ1

(MPa)

σ2 (MPa)


30 50 70 90 110

σ2 (MPa)


40 60 80 100 120 140 160

σ2 (MPa)

Kanmantoo bluestone, σ3 = 60 (MPa)

data Point

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

MSP

Table 4.8 Predicted values of failure stress by the means of each selected failure criteria for m = 36.6 and )P= 190.3 MPa

Figure 4.32 )�-)� plots, demonstrating the comparison of the selected three-dimensional failure criterion


135

As is obvious from Table 4.8 and Fig. 4.32, after adjusting the input parameter m to account

for the higher apparent strength of the cubic rock specimens, the values for the failure stress

predicted by means of three-dimensional rock failure criteria show reasonable agreement with

the true-triaxial test data. Comparison of the selected rock failure criteria revealed that the

modified simplified Priest and the Zhang-Zhu were the two criteria which predicted the rock

strength more accurately than other three-dimensional failure models (Table 4.9).

Criterion Rank σ₁₁₁₁(model)-σ₁₁₁₁(observed) (MPa)

over/underestimation (average)


(%)

Zhang-Zhu 1 -27.91 underestimation 7.00

Modified Simplified Priest 2 -16.51 underestimation 7.21

Simplified Priest 3 -44.01 underestimation 7.56


Hoek-Brown 5 -77.60 underestimation 12.13


Table 4.9 Error analysis and quantitative comparison of selected 3D failure criteria

CHAPTER 5

A case study of prediction of borehole instability

Page


5.2. Prediction of borehole instability 137

5.3. Designing the drilling fliud 143


136

5.1. Introduction

As outlined previously, a common strategy for borehole stability evaluation is to estimate the

induced stresses at the borehole vicinity by means of analytical or numerical models, and then

substitute the in situ stress components into a rock failure model to investigate whether or not

the rock failure initiates. Stress analysis based on linear elasticity theory was carried out to

estimate the induced stresses around a vertical and a deviated borehole in Chapter 3. The

existing analytical model for stress estimation around a borehole is an elastic solution, which

is a three-dimensional expansion of Kirsch equations and also is referred to as the generalised

Kirsch equations. These equations have been widely applied in the petroleum and mining

industries since they were introduced in 1962 by Hiramatsu and Oka. However, boundary

conditions on which this elastic solution was based have been inadequately addressed in the

existing literature.

Finite element analysis (FEA) was employed to first create the numerical counter part of the

analytical solution in order to clarify the boundary conditions involved in the analytical

solution. It appeared that in the case of a deviated borehole where the stress state around the

borehole is a general stress state, some simplifying assumptions were made to facilitate the

procedure of deriving the analytical solution. In other words, the three-dimensional problem

was divided into two two-dimentional problems; one on the assumption of plane strain and the

other on the assumption of anti-plane strain. It is assumed that under plane strain conditions

there is no deformation along the borehole axis and all deformations take place in planes

perpendicular to the axis of the borehole. On the other hand, under anti-plane strain conditions

it is assumed that a constant deformation along the borehole axis is the only deformation that

occurs. In order to eliminate this contradiction in the assumed boundary conditions in the

analytical model, a finite element model was developed based on a new set of boundary

conditions (Chapter 3, Section 3.5) which is in better compliance with the physics of the

problem. A detailed explanation of the proposed boundary conditions is given in Section 3.5.

After estimating stresses around the borehole, the stability of the rock material at the borehole

wall is required to be investigated. Under a given stress state, the maximum stress that can be


137

tolerated by rock is referred to as the failure tress. The stability of the rock material

surrounding the borehole can be evaluated by comparing the rock failure stress with the

maximum in situ stress at the borehole proximity. Failure stress for rock material can be

estimated by means of mathematical formulations known as failure criteria. A number of

common failure criteria were briefly explained in Chapter 2 and in Chapter 4 detailed

discussions on failure criteria which were developed especially for rock material were

presented. A new three-dimensional failure criterion was also introduced by improving the

simplified Priest criterion developed by Priest (2005).

In this chapter first the stability of a deviated borehole is evaluated and then minimum and

maximum mud weight for drilling the bore hole is calculated to demonstrate the practical

application of the techniques developed in this thesis. Some equations and relations are

repeated in the following sections for convenience.

5.2. Prediction of borehole instability

If the induced stresses around the borehole exceed the rock failure stress, failure of intact rock

is expected to occur at the borehole wall. In order to demonstrate the procedure of predicting

the borehole instability, a deviated borehole was considered in this section. The geometrical

characteristics of the deviated borehole is the same as for the case example given for FEA in

Chapter 3, i.e. a deviated borehole of radius 0.08 m with inclination of 125/10 (trend/plunge)

is considered. The borehole is assumed to be drilled in the Australian crustal rocks with

principal stresses being �0 = 45 MPa, �/ = 75 MPa and �� = 66 MPa at the depth of 3000 m.

In order to estimate the induced stresses around the deviated borehole the far-field principal

stress components measured in the global Cartesian coordinate system must be transformed

into a local Cartesian coordinate system which Z-axis coincides with the borehole axis (Fig.

3.10). Therefore, in this example the far-field stress state for the deviated bore hole can be

given by the following general stress tensor:

[ ]

−−−−−−

=9738.654476.21487.0

4476.28697.548812.13

1487.08812.131565.65

ijσ (5.2.1)


138

The associated stress transformation procedure was outlined in Chapter 3. In the case of an

unsupported deviated borehole in non-porous and isotropic material, the stresses at the

borehole wall can be calculated according to the generalised Kirsch equations, as follows:

0=rrσ

( ) ( ) θσθσσσσσθθ 2sin42cos2 xyyxyx −−−+=

0=θσ r


( )θσθσσθ sincos xzyzz −=

0=rzσ (5.2.2)

Induced stresses are at their most deviatoric state at the borehole wall and stress concentration

occurs at two opposite points on the circumference of the borehole. The angular position of

these two points of stress concentration can be estimated by applying Eq. 3.4.7. In the case of

this deviated borehole, the angular position of the two points of stress concentration are

calculated as � = 55.166° and � = 2355.166°. Therefore, induced stresses at two stress

concentration points on the wall of the unsupported deviated borehole are given by the

following stress tensor, as was calculated in Chapter 3:

[ ]

−−=

=70.8602.30

02.324.1790

000

zzzzr

zr

rzrrr

ij

σσσσσσσσσ

σ

θ

θθθθ

θ (5.2.3)

After calculating the induced stresses at two opposite points around the borehole the next step

in borehole stability evaluation is to investigate whether or not the rock material at the

borehole wall can sustain the induced stresses. It is important to remember that failure of a

particular rock type occurs when the failure stress is exceeded by the applied stresses. Since

the stress state at the borehole wall is three-dimensional, it is more suitable for the rock failure

stress to be estimated by means of a three-dimensional failure criterion. In Chapter 4 a three-

dimensional failure criterion based on the Hoek-Brown criterion was developed by modifying


139

the simplified Priest criterion. The criterion demonstrated significant accuracy in predicting

rock failure stress in three-dimensional stress state when it was examined against the nine sets

of published true-triaxial experimental data, and performed the best when compared to the

other three-dimensional rock failure criteria. According to the proposed three-dimensional

failure criterion, as also was outlined in Chapter 4, failure of intact rock occurs when:

2

123

22

23

211 2

+−+= σσσσσ HBHBf (5.2.4)

The term ��/k in Eq. 5.2.4 is calculated, for intact rock material as follows:

2

1

331 1

++=

c

HBcHBHB

m

σσσσσ (5.2.5)

Also assuming a weighting factor w ranging from 0 to 1, the term ��/k is given by:

( )10

1 323

≤≤

−+=

w

wwHB σσσ (5.2.6)

Where the weighting factor w is defined by the following expression:

5.0324.0

−=

cw

σσ

ηµ

(5.2.7)

Where parameters � and � are defined as follows:

( )

( )( )322

32

22

32

σσσσ

σσσµ

σσση

≠−

+=

−=

cc

c

m

m

(5.2.8)

It is important to remember that when �� Eq. 5.2.6 reduces to ��/k � �� and

consequently the rock failure stress, �� in Eq. 5.2.4 will be the same as the failure stress

predicted by the original two-dimensional formulation of the Hoek-Brown criterion.


140

If principal stresses acting on the borehole wall are in descending order such that �� ' �� '��, then ��, �� and �� are corresponding to major (��), intermediate (��) and minor (��)

principal stresses, respectively. It is important to note that stress components in Eq. 5.2.2 are

not principal stresses. On the other hand, the proposed failure model for assessing rock

stability has been formulated in terms of the principal stresses. Therefore, it is important to

calculate the values and directions of the principal stresses associated with the general stress

tensor in Eq. 5.2.2. The direction determined by the unit vector λ is said to be a principal

direction of the stress tensor �67 if there exists a parameter �� such that:

( ) 0=− jijpij λδσσ (5.2.9)

Where É67 is the Kronecker delta (Eq. 4.2.5). The expanded form of Eq. 5.2.9 is a set of three

linear algebraic equations which can also be expressed in the following form:

( )( )

( )0

3

2

1

=

−−

−

λλλ

σσσσσσσσσσσσ

pzzyzxz

yzpyyxy

xzxypxx

(5.2.10)

In Eq. 5.2.10 the parameters ·�, ·� and ·� are direction cosines associated with the principal

stress tensor. Furthermore, the three linear algebraic equations, given by Eq. 5.210, can be

simultaneously solved by equating the determinant of the coefficient matrix to zero, as

follows:

( )( )

( )0=

−−

−

pzzyzxz

yzpyyxy

xzxypxx

σσσσσσσσσσσσ

(5.2.11)

Evaluating the determinant in Eq. 5.2.11 gives the following characteristic equation:

0322

13 =−+− III ppp σσσ (5.2.12)


141

Where _�, _� and _� are the first, second and third invariants of the general stress tensor and are

given as follows:

zzyyxxI σσσ ++=1

( )2222 xzyzxyxxzzzzyyyyxxI σσσσσσσσσ ++−++=

( )2223 2 xzyyyzxxxyzzxzyzxyzzyyxxI σσσσσσσσσσσσ ++−+= (5.2.13)

Eq. 5.2.12 is a cubic equation so there are three distinct values of �� that provide solutions.

These roots of the cubic equation are the three principal stresses. Although the roots of the

characteristic equation can be zero or negative, they are always real (i.e. never imaginary) in

the mathematical sense. This special property means that a simple algorithm can be adopted

for solving the equation. In order to solve the equation, five further intermediate parameters

are needed to be defined as follows:

22

11 3IIJ −=

2

2792 3213

12

IIIIJ

+−=

22

313 JJJ −=

14 JJ =

=

2

3arctan31

J

Jθ (5.2.14)

The three principal stresses ��, �� and �� are given by the following expressions: (it should be

noted that �� ' �� ' ��).

3cos2 41

1θσ JI +=

( )3

32cos2 41

2

πθσ

−+=

JI


142

( )3

34cos2 41

3

πθσ

−+=

JI (5.2.15)

Therefore, principal stresses associated with the general stress tensor, given by Eq. 5.2.3 can

be calculated as:

=

=

000

06.860

00338.179

00

00

00

00

00

00

3

2

1

σσ

σ

σσ

σθθ

rr

zz (5.2.16)

Furthermore, according to the results of FEA based on the proposed boundary conditions

which were outlined in Section 3.5, the induced stress state around the deviated borehole is

given by the following stress tensor:

[ ]

−−=

=70.8685.10

85.112.1790

000

zzzzr

zr

rzrrr

ij

σσσσσσσσσ

σ

θ

θθθθ

θ (5.2.17)

The corresponding principal stress tensor for the stress tensor in Eq. 5.2.10 is given as follows:

[ ]

=

=000

070.860

0016.179

00

00

00

3

2

1

σσ

σσ ij (5.2.18)

As is obvious there is no considerable difference between the calculated stresses by the means

of the generalised Kirsch equations and the proposed numerical model. However, since the

proposed boundary conditions are in relatively better agreement with the physics of the

problem, as explained in Chapter 3, the borehole stability analysis in this section is continued

by considering the induced stress tensor given by Eq. 5.2.11.

First it is assumed that the borehole has been drilled in a granite formation, which has the

uniaxial compressive strength, �� of 229 MPa and the Hoek-Brown dilatancy parameter, m, of

29.5. Substituting the calculated values for the intermediate,�� , and minor,�� ,

principal stresses into Eqs. 5.2.4 to5.2.7, the failure stress of the rock material at the borehole


143

wall is calculated as 460.73 MPa. Comparing this value of the rock failure stress to the major

in situ principal stress, �� = 179.16 MPa, it is inferred that the rock material at the

borehole wall does not fail in compression. However, if the borehole is drilled in a rock, for

example marble, with uniaxial compressive strength of 82 MPa and the Hoek-Brown dilatancy

parameter m of 12, the failure stress calculated by Eqs. 5.2.4 to 5.2.7, is 170.6 MPa which is

smaller compared to the in situ major principal stress, �� = 179.16 MPa, and therefore failure

is anticipated to initiate at the borehole wall. Table 5.1 summarises the associated calculation

and procedure of the stability evaluation for the discussed example on the deviated borehole.

Rock α β w σ3HB σ1HB Failure stress σ1f (MPa)

Occurrence of failure at the borehole wall

Granite 77.95 84.93 0.25 21.71 467.93 460.85 (>179.16) No

Marble 11.35 12.25 0.25 21.60 188.88 170.58 (<179.16) Yes

5.3. Designing the drilling fluid

In geotechnical engineering drilling fluid is used to aid the drilling of boreholes into the Earth.

Liquid drilling fluid is often referred to as drilling mud. The three main categories of drilling

fluids are water-based mud, non-aqueous mud, known as oil-based mud, and gaseous drilling

fluid, in which a wide range of gases can be applied. The main functions of drilling fluids

include providing hydrostatic pressure to prevent the borehole wall from failing, keeping the

drill bit cool and cleaning the borehole during drilling by carrying out the drill cuttings.

Drilling mud makes a column of fluid, which exerts a radial pressure on the borehole wall.

The magnitude of this radial pressure at a depth of h is calculated as follows:

ghPw ρ= (5.3.1)

Where g is the gravitational acceleration and is usually assumes to be 9.81N/kg and � is the

density of the drilling mud, which according to Eq. 5.3.1, is expressed in kg/m3.

Table 5.1 Calculation of the failure stress for Granite and Marble


144

For an unsupported borehole the radial stress, ��, at the borehole wall is zero and, as

demonstrated in Section 5.2, the corresponding least principal stress acting on a rock element

at the borehole wall is also zero. However, due to the column of the drilling fluid the redial

pressure at the borehole wall increases to aÊ. The radial and tangential induced stresses due to

the hydrostatic pressure of the column of mud in the borehole are given as follows (Fig. 5.1):

wrr P=σ

wP−=θθσ (5.3.2)

If it is assumed that the drilling mud completely fills the borehole and the hydrostatic pressure,

aÊ is homogeneous around the circumference of the deviated borehole, then total stresses

around a deviated borehole containing a column of drilling mud (Fig. 5.1) can be calculated by

superimposing Eqs. 5.3.2 on the radial and tangential stress components (�� and ��) in Eqs.

5.2.2 as follows:

wrr P=σ

( ) ( ) wxyyxyx P−−−−+= θσθσσσσσθθ 2sin42cos2

0=θσ r


( )θσθσσθ sincos xzyzz −=

0=rzσ (5.3.3)

The hydrostatic pressure imposed on the borehole wall due to the column of drilling mud, acts

as a confining pressure and can dramatically increase the rock failure stress. In the case of

drilling a deviated borehole in a Marble formation, where the rock failure in compression is

expected to occur, it is desired to calculate the minimum hydrostatic pressure, aÊ"R6>$ to

prevent the rock material at the borehole wall from failure.


145

Superimposing Eq. 5.3.2 on the stress tensor calculated by the means of the finite element

analysis, based on the proposed boundary conditions, the general stress state around the

deviated borehole is given as:

[ ]

−−−=

−+

=70.8685.10

85.112.1790

00

w

w

zzzzr

zwr

rzrwrr

ij P

P

P

P

σσσσσσσσσ

σ

θ

θθθθ

θ (5.3.4)

The principal stress tensor associated with this general stress tensor can be calculated by

applying Eqs. 5.2.13 to 5.2.15. Substituting the principal stresses into the proposed strength

model given by Eqs. 5.2.4 to 5.2.7 and equating the calculated rock failure stress given by

Eq.5.2.4 to the in situ major principal stress, i.e. �� 179.12�aÊ, the minimum

hydrostatic pressure required for preventing the failure of the rock material at the borehole

wall is calculated as aÊ"R6>$ = 176.9 MPa. Therefore, according to Eq.5.3.1 the minimum

density of the drilling mud for safely drilling the deviated borehole at the depth of 3000 m is

calculated as:

3

26

min 873.6010300081.9

109.176

mKg

mkgN

mN

=×

×=ρ

The upper limit or the maximum allowable hydrostatic pressure in the borehole aÊ"Rb�$ is

constrained by the hydraulic fracturing stress. Hydraulic fracture is a tensile fracture which is

assumed to occur if any of stress components at the borehole wall becomes sufficiently tensile

��

�� aÊ

�� aÊ

Figure 5.1 Principal in situ stresses acting on a rock element at the borehole wall, with drilling fluid


146

to overcome the rock tensile strength, Ëi. Since the most negative stress that can exist at the

borehole wall at the two points of stress concentration around the borehole circumference, is

the tangential stress, the condition for hydraulic fracturing is given as follows:

0TPw −=−θθσ (5.3.5)

Where Ëi is the rock tensile strength. Therefore the maximum allowable hydrostatic pressure

in the borehole is calculated as follows:

0(max) TPw += θθσ (5.3.6)

The tangential stress, �� in Eq. 5.3.6 can be calculated for �=55.166° using the generalised

Kirsch equations. However, in this study the results of the FEA are applied for determining the

tangential stress component around the deviated borehole.

Furthermore, Eqs. 5.2.4 to 5.2.7 can be applied to calculate the tensile strength of the rock, Ëi

through the following procedure:

1 - Equating ��/k to zero and calculate the parameter ��/k as follows:

( )

+−= 2

12

3 42

mmcHB

σσ (5.3.7)

2- Calculate parameters 3 and 4, by equating the intermediate principal stress to zero as

follows:

23

2

3 σσµ

σση cc and

m =−

= (5.3.8)

3- Calculate the weighting factor w by substituting parameters 3 and 4 from Eq. 5.3.8 into

Eq. 5.2.7.

4- Setting �� in Eq. 5.2.4 to zero, which yields the following equation:


147

024.012 233

2

13 =−

−− σσ

σσ

ηµ

c (5.3.9)

5- Solving Eq. 5.3.9 for �� and setting 30 σ−=T

Following the five-step solution, the tensile strength of marble into which the deviated

borehole has been drilled is calculated as Ëi = 4.77 MPa. Therefore, considering Eq. 5.3.4 and

Eq. 5.3.6 the maximum mud pressure is calculated as:

MPaTPw 9.18377.412.1790(max) =+=+= θθσ

Consequently, considering Eq. 5.3.1 the maximum allowable density of the drilling fluid is

calculated as:

3

26

max 726.6248300081.9

109.183

mKg

mKgN

mN

=×

×=ρ

CHAPTER 6

Conclusion

Page

6.1. Conclusion 147

6.2. Recommendations for future studies 148


147

6.1. Summary and conclusions

Finite element analysis was carried out for calculating the induced stresses around a borehole

in a continuum, homogeneous, isotropic and linearly elastic rock. Results from the FEA and

the analytical models, i.e. the generalised Kirsch equations, were cross-evaluated for vertical

and deviated boreholes to clarify the boundary conditions involved in deriving the existing

analytical model. It appeared, however, that the boundary conditions assumed for deriving the

generalised Kirsch equations are incompatible with the real life situation in the physical sense,

i.e. in reality deformation of a continuum body cannot be assumed to be in plane strain and in

anti-plane strain states simultaneously. The detailed explanation of the reason for this

incompatibility is given in Chapter 3.

In order to address the contradictory boundary conditions assumed in the existing analytical

model, a finite element analysis was carried out by applying a new set of boundary conditions.

Assuming displacement as the unknown variable, the proposed boundary conditions can be

given by the following strain tensor: (as is also given by Eq. 3.5.3 and is repeated here for

convenience)

[ ]

∂∂

∂∂

∂∂

∂∂

∂∂

+∂

∂

∂∂

∂∂

+∂

∂∂

∂

=

0z

u

z

u

z

u

y

u

x

u

y

u

z

u

x

u

y

u

x

u

yx

yyyx

xyxx

ε (6.1.1)

The underlying assumption for proposing this set of boundary conditions is that displacements

along the borehole axis are constrained by nearby geo-materials and are negligible, similar to

the plane strain conditions in the physical sense. However, since the out-of-plane shear

components appear in the corresponding stress tensor (Eq. 3.5.4), the problem cannot be

considered as a plane strain problem.


148

Results from the FEA revealed that under the proposed boundary conditions, no significant

changes occur in the values of radial, ��, tangential, ��, vertical, �� and in-plane shear, ��,

stresses around the borehole when compared to the generalised Kirsch equations. However

calculated values for out-of plane shear stresses, �� and ��, given by Eq. 3.3.11,

demonstrated a dramatic change. Based on the results of the FEA the formulations of the out-

of-plane shear stress components were modified, as was out lined in Chapter 3, Eqs. 3.5.6.

These equations are also presented here for convenience:

( )

+−=

2

2

1sincos2

1

r

axzyzz θσθσσθ

( )

−+=

2

2

1cossin2r

axzyzrz θσθσσ (6.1.2)

A comprehensive study was also conducted on three-dimensional rock failure criteria. A new

three-dimensional failure criterion was developed by modifying the simplified Priest criterion

as was outlined in Chapter 4, Section 4.7.2. True-triaxial experiments were conducted at the

University of Adelaide and the results of the true-triaxial testing along with nine sets of

published true-triaxial experimental data were applied to evaluate the proposed criterion

versus other selected three-dimensional rock failure criteria which are more commonly applied

in rock mechanics studies. Comparison of the selected three-dimensional rock failure criteria

with the true-triaxial experimental data revealed that the proposed criterion, i.e. the modified

simplified Priest criterion, can evaluate the rock strength under three-dimensional stress more

accurately than other criteria in most cases. In some cases the Zhang-Zhu criterion provides

more accurate prediction of the rock failure stress, however, even in such cases the Zhang-Zhu

and the modified simplified Priest criteria are significantly close.

6.2. Recommendations for future studies

Although a numerical solution was presented in this study for implementing the proposed

boundary conditions to calculate values of out-of-plane shear components ( �� and ��)

around a deviated borehole, it merits developing an analytical solution for calculating the out-


149

of plane shears in order to analytically prove the formulation proposed for calculating the

longitudinal stresses (�� and ��) around a deviated borehole. Furthermore, this finite element

model can be utilised as a platform for developing more sophisticated models for stress

analysis around an opening excavated in geo-materials with more complicated constitutive

behaviour and geological features.

The rock failure criterion for predicting the failure stress of intact rock developed in Chapter 4

can be generalised in order to evaluate the strength of rock masses when estimating the

stability of underground excavations with lager cross sections than of a borehole.

Furthermore, a five-step solution for calculating the tensile strength of rock was outlined in

this study by applying the proposed three-dimensional failure model. It would be worthwhile

conducting a series of uniaxial tensile tests to evaluate the accuracy of the proposed method in

predicting the tensile strength of rock.

150

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Appendix A

The Finite Element Method

APPENDIX A The finite element method

155

The Finite Element Method (FEM)

The finite element method (FEM) is a matrix algebraic method developed to solve partial

differential equations (PDEs) in order to calculate linear or non-linear response of continuous

physical systems to applied boundary conditions. The basic principle of the FEM is the

division of this physical system into a number of simply shaped continuous sub-domains

(finite elements) in order to find approximate solutions for the PDEs. This discretisation may

facilitate the solution of a PDE system at discrete points in the model space in cases where an

analytical solution is impossible to obtain. In order to provide a unique solution and

convergence of the system of equations, appropriate boundary conditions have to be defined.

The simple shape of each finite element is defined by its corner nodes which may be shared

with adjacent elements, depending on their spatial position in the model space. All nodes and

elements are uniquely defined in the model space by their identifier. The entirety of all entities

defines the finite element mesh which is the discretised representation of the continuous

problem space.

Once the model space is discretised, a set of linear equations of the unknown field variables is

approximated for each element. The most common approach for problems in continuum

mechanics is the displacement method in which the nodal displacement 8!�Ì"#$: is the unknown

field variable. Solving the problem for the unknown field variable requires that the equilibrium

of forces, internal continuity and a constitutive relationship with respect to the material

behaviour be satisfied. Through these constitutive laws stresses are related to strains and hence

to nodal displacements. The equilibrium equation can be derived from the equation of motion

for small movements which can be written (in index notation) as:

iij

ijaB

xρρ

σ=+

∂∂

(A.1)

Where � is the density, U6 is the body force and C6 is the acceleration. For the elastic stress

analysis around a borehole, acceleration can be assumed to be negligible since there exist no

instantaneous displacements at the borehole wall and the equations of static equilibrium are

applicable. In the absence of body forces, the resultant of all surface forces as well as the


156

resultant moment about any axis has to be vanished in the state of equilibrium. Therefore the

equilibrium equation can be written as:

0=∂∂

j

ij

x

σ (A.2)

For perfect linear elasticity, the strain tensor (Í� is proportional to the stress tensor �67. This

relationship is termed as Hook’s law and can be expressed as:

klijklij C εσ = (A.3)

The forth-order tensor n67Í�, which is a 9×9 matrix, represents the elasticity tensor containing

81 components. However, the number of components reduces to 36 since both the stress tensor

(�67) and the strain tensor ((Í�) are symmetric, having only six distinct components.

Substituting Eq. A.3 into Eq. A2, the equilibrium equation of forces can be written as:

[ ] 021 =

∂∂

+∂∂

∂∂=

∂∂

i

j

j

iijkl

jklijkl

j x

u

x

uC

xC

xε (A.4)

After each element has been approximated into a set of linear equations, the entirety of these

approximation functions are assembled into a global equation of motion, which enables the

solution of the numerical problem.

On the other hand, the matrix algebraic formulation of the fundamental equation of motion can

be expressed as:

uKt

uC

t

uMF

vvv

v+

∂∂+

∂∂=

2

2

(A.5)

Where ÎÏ is the mass matrix, nÐ is the damping matrix and ÑÏ is the stiffness matrix. In this

study, the finite element analysis (FEA) is restricted to static and Cauchy static problems,

neglecting accelerations GÒK��ÌÒ KM and velocities GÒ��Ì

Ò M. Therefore Eq. A.5 is reduced to the static

equilibrium equation as follows:


157

uKFvv

= (A.6)

Since for isotropic materials the elasticity matrix, C in Eq. A.3, is the same as the stiffness

matrix, K in Eq. A.6, the static equilibrium equation given by Eq. A.6 can be inverted to solve

for the unknown displacements once the stiffness matrix is known. Subsequently, the stress

and strain tensors can be derived.

Two basic principles, the Eulerian and the Lagrangian formulations can be followed to solve

the constitutive equations within the finite element mesh. In the Eulerian approach, material

properties or field variables migrate through the finite element mesh which is not deforming

during the analysis. This prevents numerical instabilities due to excessive distortion of the

finite element mesh. However, because the continuum is not explicitly defined, boundary

conditions and boundary layers are difficult to trace or redefine. In the Lagrangian approach

the material properties are explicitly defined for each element and the finite element mesh

deforms during the analysis. The disadvantage of the Lagrangian approach is the explicit

definition of the continuum, which leads to excessive distortion of the finite element mesh and

to numerical instability when addressing large deformation problems. Since in the linear

elastic solution deformations around a borehole are assumed to be infinitesimal, the

ABAQUS/Standard implementation of the Lagrangian approach was adopted in this study.

The focus of this thesis is to apply the FEM for stress analysis around a borehole drilled in an

isotropic, homogeneous and linearly elastic rock. To provide a description of the complete

mathematical background of the FEM is beyond the scope of this study. Hence, only the

fundamental concept and the basic equations have been outlined. For a complete mathematical

description of the FEM the reader is referred to Zienkiewicz and Taylor (1994) and

Zienkiewicz et al. (2005).

Mesh resolution

The accuracy of the FEA is directly dependent on the discretisation resolution. The behaviour

of the discretised continuum, for example, the field variable !�Ì"#$ is described by a set of

linear equations such as !�ÌÓ � 3i [ 3�!�Ì, in which 3 indicates the approximation coefficients.

Therefore, a finite discretisation is required in model regions where high gradients of !�Ì"#$


158

occur, for example, in the regions adjacent to the borehole wall. Fig. 3.4 demonstrates the

impact of discretisation resolution on the accuracy of numerical approximation to a non-linear

function.

(c) u(X) (b) u(X) Numerical

approximation

Numerical error

(a) u(X)

Figure A.1 The numerical error of the observed field variable (in this case Ô��Ì"q$) can be minimized by increasing the discretisation resolution stepwise from (a) to (c).

APPENDIX B

Qunatitative comparison between analytical and numerical

models

159

θ (deg)

Analytical solution Numerical Analysis Error calculation (%)

σrr σθθ σaa σrθ σra σθa σrr σθθ σaa σrθ σra σθa σrr (e) σθθ (e) σaa (e) σrθ (e) σra (e) σθa (e) 1.5 9.69 163.39 84.58 0.33 0.00 0.00 9.52 167.04 85.79 0.32 0.00 0.00 1.74 2.23 1.44 1.81 0.00 0.00 4.5 9.66 162.84 84.37 0.98 0.00 0.00 9.49 166.46 85.58 0.95 0.00 0.00 1.74 2.22 1.43 2.72 0.00 0.00 7.5 9.59 161.74 83.97 1.62 0.00 0.00 9.43 165.31 85.16 1.59 0.00 0.00 1.75 2.20 1.42 1.65 0.00 0.00

10.5 9.50 160.12 83.37 2.24 0.00 0.00 9.33 163.60 84.53 2.20 0.00 0.00 1.76 2.17 1.39 2.02 0.00 0.00 13.5 9.38 157.98 82.57 2.84 0.00 0.00 9.21 161.35 83.70 2.79 0.00 0.00 1.77 2.13 1.36 1.82 0.00 0.00 16.5 9.23 155.34 81.60 3.41 0.00 0.00 9.07 158.58 82.68 3.36 0.00 0.00 1.78 2.08 1.32 1.64 0.00 0.00 19.5 9.06 152.25 80.46 3.94 0.00 0.00 8.89 155.33 81.48 3.87 0.00 0.00 1.80 2.02 1.27 1.95 0.00 0.00 22.5 8.86 148.72 79.15 4.43 0.00 0.00 8.70 151.63 80.11 4.34 0.00 0.00 1.81 1.95 1.21 2.14 0.00 0.00 25.5 8.64 144.81 77.71 4.87 0.00 0.00 8.48 147.51 78.60 4.76 0.00 0.00 1.82 1.87 1.15 2.23 0.00 0.00 28.5 8.40 140.55 76.13 5.25 0.00 0.00 8.24 143.04 76.95 5.13 0.00 0.00 1.82 1.77 1.07 2.27 0.00 0.00 31.5 8.14 135.99 74.45 5.58 0.00 0.00 7.99 138.25 75.18 5.46 0.00 0.00 1.82 1.66 0.99 2.26 0.00 0.00 34.5 7.87 131.18 72.67 5.85 0.00 0.00 7.73 133.20 73.32 5.72 0.00 0.00 1.81 1.54 0.90 2.21 0.00 0.00 37.5 7.59 126.17 70.81 6.05 0.00 0.00 7.45 127.94 71.38 5.92 0.00 0.00 1.79 1.40 0.81 2.13 0.00 0.00 40.5 7.30 121.02 68.91 6.19 0.00 0.00 7.17 122.53 69.39 6.06 0.00 0.00 1.76 1.25 0.70 2.03 0.00 0.00 43.5 7.00 115.78 66.97 6.26 0.00 0.00 6.88 117.03 67.37 6.14 0.00 0.00 1.72 1.08 0.59 1.90 0.00 0.00 46.5 6.70 110.52 65.03 6.26 0.00 0.00 6.59 111.50 65.33 6.18 0.00 0.00 1.66 0.89 0.47 1.14 0.00 0.00 49.5 6.41 105.28 63.09 6.19 0.00 0.00 6.31 106.00 63.31 6.09 0.00 0.00 1.58 0.69 0.34 1.58 0.00 0.00 52.5 6.12 100.13 61.19 6.05 0.00 0.00 6.03 100.59 61.32 5.97 0.00 0.00 1.49 0.46 0.21 1.39 0.00 0.00 55.5 5.83 95.12 59.33 5.85 0.00 0.00 5.75 95.34 59.38 5.68 0.00 0.00 1.38 0.23 0.08 2.89 0.00 0.00 58.5 5.56 90.31 57.55 5.58 0.00 0.00 5.49 90.29 57.52 5.43 0.00 0.00 1.25 0.02 0.05 2.73 0.00 0.00 61.5 5.31 85.75 55.87 5.25 0.00 0.00 5.25 85.50 55.76 5.12 0.00 0.00 1.11 0.29 0.19 2.56 0.00 0.00 64.5 5.06 81.49 54.29 4.87 0.00 0.00 5.02 81.03 54.12 4.75 0.00 0.00 0.95 0.56 0.33 2.39 0.00 0.00 67.5 4.84 77.57 52.85 4.43 0.00 0.00 4.81 76.93 52.61 4.33 0.00 0.00 0.79 0.84 0.45 2.21 0.00 0.00 70.5 4.64 74.05 51.54 3.94 0.00 0.00 4.62 73.23 51.25 3.86 0.00 0.00 0.63 1.11 0.58 2.00 0.00 0.00 73.5 4.47 70.96 50.40 3.41 0.00 0.00 4.45 69.98 50.05 3.35 0.00 0.00 0.47 1.37 0.69 1.75 0.00 0.00 76.5 4.32 68.32 49.43 2.84 0.00 0.00 4.31 67.22 49.03 2.80 0.00 0.00 0.32 1.61 0.79 1.43 0.00 0.00 79.5 4.20 66.18 48.63 2.24 0.00 0.00 4.19 64.97 48.21 2.22 0.00 0.00 0.20 1.82 0.87 0.97 0.00 0.00 82.5 4.11 64.55 48.03 1.62 0.00 0.00 4.10 63.27 47.58 1.62 0.00 0.00 0.10 1.99 0.94 0.17 0.00 0.00 85.5 4.05 63.46 47.63 0.98 0.00 0.00 4.05 62.12 47.16 1.00 0.00 0.00 0.04 2.11 0.98 1.68 0.00 0.00 88.5 4.02 62.91 47.42 0.33 0.00 0.00 4.02 61.54 46.95 0.33 0.00 0.00 0.03 2.17 1.01 1.78 0.00 0.00

Table B.1 Error analysis of the finite element model in comparison with the analytical solution, for calculating the induced stresses around the vertical borehole (for a quarter-model)

160

θ (deg)

Analytical solution Numerical Analysis Error calculation (%)

σrr σθθ σaa σrθ σra σθa σrr σθθ σaa σrθ σra σθa σrr (e) σθθ (e) σaa (e) σrθ (e) σra (e) σθa (e) 1.5 6.02 98.38 60.51 5.90 -0.02 -4.61 5.92 98.71 60.59 5.63 -0.02 -4.77 1.60 0.34 0.14 4.54 1.26 3.49 4.5 6.30 103.42 62.37 6.06 -0.04 -4.58 6.19 103.97 62.52 5.78 -0.04 -4.74 1.71 0.53 0.25 4.64 0.79 3.47 7.5 6.59 108.56 64.27 6.16 -0.05 -4.54 6.47 109.34 64.50 5.86 -0.05 -4.70 1.80 0.72 0.36 4.74 1.74 3.46

10.5 6.89 113.75 66.19 6.18 -0.07 -4.49 6.76 114.77 66.50 5.88 -0.07 -4.64 1.88 0.89 0.47 4.85 2.30 3.43 13.5 7.18 118.94 68.11 6.14 -0.08 -4.42 7.04 120.19 68.49 5.84 -0.08 -4.57 1.93 1.05 0.57 4.96 2.67 3.40 16.5 7.47 124.06 70.00 6.03 -0.10 -4.35 7.32 125.55 70.47 5.83 -0.09 -4.49 1.97 1.19 0.67 3.41 2.94 3.36 19.5 7.75 129.07 71.85 5.86 -0.11 -4.26 7.59 130.78 72.40 5.65 -0.11 -4.40 2.00 1.33 0.76 3.46 3.15 3.31 22.5 8.02 133.90 73.64 5.62 -0.12 -4.16 7.86 135.84 74.26 5.42 -0.12 -4.29 2.01 1.45 0.85 3.49 3.31 3.26 25.5 8.28 138.50 75.34 5.32 -0.14 -4.05 8.12 140.66 76.03 5.13 -0.13 -4.18 2.01 1.56 0.92 3.49 3.44 3.19 28.5 8.53 142.82 76.94 4.96 -0.15 -3.92 8.35 145.19 77.70 4.79 -0.14 -4.05 2.00 1.65 1.00 3.44 3.56 3.12 31.5 8.75 146.82 78.42 4.55 -0.16 -3.79 8.58 149.38 79.25 4.39 -0.15 -3.90 1.99 1.74 1.06 3.34 3.66 3.03 34.5 8.96 150.46 79.76 4.08 -0.17 -3.65 8.78 153.19 80.65 3.95 -0.17 -3.75 1.98 1.82 1.12 3.16 3.75 2.93 37.5 9.14 153.68 80.95 3.58 -0.18 -3.49 8.96 156.57 81.90 3.47 -0.18 -3.59 1.96 1.88 1.17 2.86 3.83 2.81 40.5 9.30 156.46 81.98 3.03 -0.19 -3.33 9.11 159.49 82.97 2.96 -0.19 -3.42 1.94 1.93 1.22 2.39 3.89 2.67 43.5 9.43 158.76 82.83 2.45 -0.20 -3.16 9.24 161.91 83.87 2.37 -0.20 -3.23 1.93 1.98 1.25 3.25 3.96 2.51 46.5 9.53 160.57 83.50 1.84 -0.21 -2.97 9.35 163.81 84.57 1.77 -0.21 -3.04 1.91 2.02 1.28 4.03 4.01 2.32 49.5 9.60 161.86 83.97 1.21 -0.22 -2.78 9.42 165.16 85.07 1.16 -0.21 -2.84 1.90 2.04 1.30 4.70 4.06 2.09 52.5 9.64 162.61 84.25 0.57 -0.23 -2.59 9.46 165.96 85.36 0.56 -0.22 -2.65 1.89 2.06 1.31 3.26 4.11 2.59 55.5 9.65 162.82 84.33 -0.07 -0.24 -2.38 9.47 166.19 85.45 -0.07 -0.23 -2.46 1.88 2.07 1.32 4.51 4.15 3.17 58.5 9.64 162.49 84.21 -0.72 -0.25 -2.17 9.45 165.85 85.32 -0.69 -0.24 -2.26 1.88 2.07 1.32 4.19 4.19 3.84 61.5 9.59 161.62 83.89 -1.36 -0.25 -1.96 9.41 164.94 84.99 -1.30 -0.24 -2.02 1.88 2.06 1.31 4.15 4.22 3.13 64.5 9.51 160.21 83.37 -1.98 -0.26 -1.73 9.33 163.48 84.45 -1.88 -0.25 -1.79 1.88 2.04 1.30 4.75 4.25 3.38 67.5 9.40 158.29 82.66 -2.58 -0.26 -1.51 9.22 161.48 83.71 -2.49 -0.25 -1.56 1.89 2.01 1.27 3.43 4.28 3.70 70.5 9.26 155.88 81.76 -3.15 -0.27 -1.28 9.09 158.96 82.78 -3.00 -0.26 -1.31 1.90 1.97 1.24 4.87 4.30 2.56 73.5 9.10 153.00 80.70 -3.69 -0.27 -1.04 8.93 155.95 81.67 -3.54 -0.26 -1.08 1.91 1.93 1.20 4.05 4.32 3.78 76.5 8.91 149.68 79.47 -4.19 -0.28 -0.80 8.74 152.48 80.39 -3.99 -0.26 -0.83 1.91 1.87 1.16 4.89 4.33 3.21 79.5 8.70 145.96 78.10 -4.64 -0.28 -0.57 8.54 148.60 78.96 -4.48 -0.27 -0.58 1.92 1.80 1.10 3.58 4.35 3.05 82.5 8.47 141.89 76.59 -5.04 -0.28 -0.32 8.31 144.34 77.39 -4.86 -0.27 -0.33 1.92 1.73 1.04 3.64 4.36 2.63 85.5 8.23 137.50 74.97 -5.39 -0.28 -0.08 8.07 139.75 75.70 -5.19 -0.27 -0.08 1.91 1.64 0.98 3.65 4.36 3.37 88.5 7.96 132.84 73.24 -5.68 -0.28 0.16 7.81 134.88 73.91 -5.47 -0.27 0.17 1.90 1.54 0.90 3.62 4.37 3.75

Table B.2 Error analysis of the finite element model in comparison with the analytical solution (the generalised Kirsch equations), for calculating the induced stresses around a deviated borehole (for a quarter-model)

161

θ (deg)

Numerical Analysis Analytical solution Error calculation (%)

σrr σθθ σaa σrθ σra σθa σrr σθθ σaa σrθ σra σθa σrr (e) σθθ (e) σaa (e) σrθ (e) σra (e) σθa (e) 1.5 5.92 98.68 60.31 -5.71 -0.05 -3.29 6.02 98.38 60.51 -5.90 -0.02 -4.61 1.64 0.30 0.33 3.32 0.30 0.33 4.5 6.19 103.91 62.07 -5.86 -0.08 -3.27 6.30 103.42 62.37 -6.06 -0.04 -4.58 1.84 0.48 0.48 3.39 0.48 0.48 7.5 6.46 109.26 63.88 -5.94 -0.11 -3.25 6.59 108.56 64.27 -6.16 -0.05 -4.54 2.02 0.65 0.61 3.45 0.65 0.61

10.5 6.74 114.67 65.71 -5.96 -0.14 -3.21 6.89 113.75 66.19 -6.18 -0.07 -4.49 2.16 0.80 0.72 3.51 0.80 0.72 13.5 7.02 120.07 67.54 -5.92 -0.17 -3.16 7.18 118.94 68.11 -6.14 -0.08 -4.42 2.29 0.95 0.83 3.55 0.95 0.83 16.5 7.29 125.41 69.35 -5.82 -0.20 -3.11 7.47 124.06 70.00 -6.03 -0.10 -4.35 2.39 1.08 0.93 3.57 1.08 0.93 19.5 7.56 130.62 71.12 -5.65 -0.22 -3.05 7.75 129.07 71.85 -5.86 -0.11 -4.26 2.48 1.21 1.02 3.58 1.21 1.02 22.5 7.82 135.66 72.83 -5.42 -0.25 -2.97 8.02 133.90 73.64 -5.62 -0.12 -4.16 2.55 1.32 1.10 3.56 1.32 1.10 25.5 8.07 140.46 74.45 -5.13 -0.28 -2.89 8.28 138.50 75.34 -5.32 -0.14 -4.05 2.60 1.42 1.18 3.50 1.42 1.18 28.5 8.31 144.97 75.97 -4.79 -0.30 -2.81 8.53 142.82 76.94 -4.96 -0.15 -3.92 2.65 1.50 1.26 3.39 1.50 1.26 31.5 8.52 149.15 77.37 -4.40 -0.33 -2.71 8.75 146.82 78.42 -4.55 -0.16 -3.79 2.68 1.58 1.33 3.23 1.58 1.33 34.5 8.72 152.94 78.64 -3.96 -0.35 -2.61 8.96 150.46 79.76 -4.08 -0.17 -3.65 2.71 1.65 1.41 2.97 1.65 1.41 37.5 8.89 156.30 79.75 -3.48 -0.37 -2.50 9.14 153.68 80.95 -3.58 -0.18 -3.49 2.74 1.71 1.48 2.60 1.71 1.48 40.5 9.05 159.21 80.70 -2.97 -0.39 -2.38 9.30 156.46 81.98 -3.03 -0.19 -3.33 2.76 1.75 1.56 2.02 1.75 1.56 43.5 9.17 161.61 81.47 -2.38 -0.41 -2.26 9.43 158.76 82.83 -2.45 -0.20 -3.16 2.78 1.79 1.64 2.73 1.79 1.64 46.5 9.27 163.50 82.06 -1.79 -0.43 -2.13 9.53 160.57 83.50 -1.84 -0.21 -2.97 2.80 1.82 1.73 2.74 1.82 1.73 49.5 9.34 164.84 82.45 -1.16 -0.45 -1.99 9.60 161.86 83.97 -1.21 -0.22 -2.78 2.82 1.84 1.81 4.31 1.84 1.81 52.5 9.37 165.63 82.64 -0.56 -0.47 -1.85 9.64 162.61 84.25 -0.57 -0.23 -2.59 2.85 1.85 1.91 2.29 1.85 1.91 55.5 9.38 165.85 82.64 0.07 -0.48 -1.71 9.65 162.82 84.33 0.07 -0.24 -2.38 2.88 1.86 2.01 4.30 1.86 2.01 58.5 9.36 165.50 82.43 0.69 -0.50 -1.55 9.64 162.49 84.21 0.72 -0.25 -2.17 2.91 1.85 2.11 3.52 1.85 2.11 61.5 9.31 164.58 82.02 1.30 -0.51 -1.40 9.59 161.62 83.89 1.36 -0.25 -1.96 2.95 1.84 2.23 3.81 1.84 2.23 64.5 9.23 163.12 81.41 1.90 -0.52 -1.24 9.51 160.21 83.37 1.98 -0.26 -1.73 2.99 1.81 2.35 4.04 1.81 2.35 67.5 9.12 161.11 80.61 2.49 -0.53 -1.08 9.40 158.29 82.66 2.58 -0.26 -1.51 3.03 1.78 2.47 3.65 1.78 2.47 70.5 8.99 158.59 79.63 3.00 -0.54 -0.91 9.26 155.88 81.76 3.15 -0.27 -1.28 3.07 1.74 2.61 4.94 1.74 2.61 73.5 8.83 155.58 78.48 3.56 -0.55 -0.74 9.10 153.00 80.70 3.69 -0.27 -1.04 3.11 1.68 2.75 3.64 1.68 2.75 76.5 8.64 152.11 77.17 3.99 -0.55 -0.57 8.91 149.68 79.47 4.19 -0.28 -0.80 3.16 1.62 2.89 4.76 1.62 2.89 79.5 8.43 148.23 75.72 4.46 -0.56 -0.40 8.70 145.96 78.10 4.64 -0.28 -0.57 3.19 1.55 3.05 3.89 1.55 3.05 82.5 8.21 143.97 74.13 4.85 -0.56 -0.23 8.47 141.89 76.59 5.04 -0.28 -0.32 3.23 1.47 3.21 3.91 1.47 3.21 85.5 7.97 139.38 72.44 5.18 -0.56 -0.06 8.23 137.50 74.97 5.39 -0.28 -0.08 3.26 1.37 3.38 3.90 1.37 3.38 88.5 7.71 134.52 70.65 5.46 -0.56 0.12 7.96 132.84 73.24 5.68 -0.28 0.16 3.27 1.27 3.55 3.86 1.27 3.55

Table B.3 Error analysis of the finite element analysis based on the proposed boundary conditions in comparison with the analytical solution (the generalised Kirsch’s equations), for calculating the induced stresses around a deviated borehole (for a quarter-model)

162

r (m) Numerical analysis Analytical solution Error calculation (%)

σrr σθθ σaa σrθ σra σaθ σrr σθθ σaa σrθ σra σaθ σrr (e) σθθ (e) σaa (e) σrθ (e) σra (e) σθa (e) 0.08 0.00 180.12 86.71 0.00 0.00 -3.00 0.00 179.24 86.70 0.00 0.00 -3.02 0.00 0.49 0.02 0.00 0.00 0.60 0.085 9.46 165.96 85.36 0.08 -0.25 -2.79 9.65 162.82 84.33 0.08 -0.24 -2.85 2.01 1.93 1.22 4.99 2.69 2.01 0.095 22.51 142.04 81.55 0.17 -0.63 -2.51 22.31 139.71 80.67 0.16 -0.61 -2.58 0.88 1.67 1.10 4.13 2.66 2.54 0.105 30.14 126.49 78.78 0.21 -0.91 -2.32 29.78 124.62 78.00 0.20 -0.88 -2.39 1.20 1.50 1.00 5.74 2.94 2.59 0.115 34.83 115.85 76.70 0.23 -1.11 -2.20 34.42 114.26 76.00 0.22 -1.08 -2.24 1.20 1.39 0.92 5.08 2.56 1.93 0.125 37.85 108.27 75.11 0.24 -1.27 -2.09 37.43 106.85 74.46 0.23 -1.24 -2.13 1.12 1.33 0.86 4.75 2.38 1.71 0.135 39.86 102.67 73.85 0.24 -1.40 -2.00 39.45 101.37 73.25 0.23 -1.36 -2.04 1.04 1.28 0.82 4.56 2.37 2.00 0.145 41.24 98.42 72.84 0.24 -1.50 -1.92 40.85 97.20 72.28 0.23 -1.46 -1.97 0.95 1.25 0.78 4.44 2.67 2.64 0.155 42.21 95.11 72.03 0.24 -1.58 -1.86 41.85 93.95 71.49 0.23 -1.54 -1.91 0.87 1.23 0.75 4.34 2.62 2.81 0.165 42.91 92.49 71.35 0.23 -1.64 -1.81 42.57 91.38 70.85 0.23 -1.61 -1.86 0.80 1.22 0.72 4.26 2.08 2.73 0.175 43.42 90.37 70.79 0.23 -1.71 -1.78 43.10 89.30 70.30 0.22 -1.66 -1.82 0.74 1.21 0.70 4.19 2.70 2.20 0.185 43.81 88.64 70.32 0.23 -1.74 -1.74 43.51 87.59 69.85 0.22 -1.71 -1.79 0.69 1.20 0.67 4.11 2.07 2.78 0.195 44.10 87.20 69.92 0.22 -1.79 -1.72 43.82 86.17 69.46 0.22 -1.75 -1.76 0.64 1.19 0.66 4.03 2.61 2.26 0.205 44.32 85.99 69.57 0.22 -1.82 -1.69 44.06 84.99 69.13 0.21 -1.78 -1.74 0.59 1.18 0.64 3.94 2.02 2.69 0.215 44.49 84.96 69.27 0.22 -1.85 -1.68 44.25 83.98 68.84 0.21 -1.81 -1.72 0.55 1.17 0.62 3.85 2.02 2.46 0.225 44.63 84.08 69.01 0.22 -1.88 -1.65 44.40 83.11 68.59 0.21 -1.84 -1.70 0.52 1.16 0.61 3.75 2.60 2.78 0.235 44.74 83.32 68.78 0.21 -1.91 -1.64 44.52 82.37 68.38 0.21 -1.86 -1.68 0.49 1.16 0.60 3.65 2.65 2.62 0.245 44.82 82.66 68.58 0.21 -1.93 -1.63 44.62 81.72 68.18 0.20 -1.88 -1.67 0.46 1.15 0.59 3.54 2.68 2.41 0.255 44.89 82.08 68.40 0.21 -1.94 -1.61 44.70 81.15 68.01 0.20 -1.89 -1.66 0.43 1.14 0.57 3.42 2.63 2.65 0.265 44.95 81.56 68.24 0.21 -1.96 -1.61 44.77 80.65 67.86 0.20 -1.91 -1.65 0.41 1.13 0.56 3.30 2.46 2.32 0.275 45.00 81.11 68.10 0.20 -1.96 -1.59 44.83 80.21 67.73 0.20 -1.92 -1.64 0.39 1.12 0.55 3.18 2.10 2.62 0.285 45.04 80.70 67.97 0.20 -1.98 -1.59 44.87 79.82 67.61 0.20 -1.94 -1.63 0.37 1.10 0.54 3.05 2.29 2.34 0.295 45.07 80.34 67.86 0.20 -2.00 -1.59 44.91 79.47 67.50 0.20 -1.95 -1.62 0.35 1.09 0.53 2.92 2.51 2.07 0.305 45.09 80.01 67.75 0.20 -2.00 -1.57 44.95 79.15 67.40 0.19 -1.96 -1.61 0.33 1.08 0.52 2.78 2.23 2.44 0.315 45.12 79.71 67.65 0.20 -2.02 -1.56 44.97 78.87 67.31 0.19 -1.97 -1.61 0.31 1.07 0.51 2.65 2.48 2.70 0.325 45.13 79.44 67.57 0.20 -2.02 -1.56 45.00 78.62 67.23 0.19 -1.97 -1.60 0.30 1.05 0.50 2.50 2.24 2.61 0.335 45.15 79.19 67.49 0.20 -2.03 -1.55 45.02 78.38 67.16 0.19 -1.98 -1.60 0.29 1.04 0.49 2.36 2.23 2.85 0.345 45.16 78.97 67.41 0.19 -2.04 -1.55 45.04 78.17 67.09 0.19 -1.99 -1.59 0.27 1.02 0.48 2.21 2.40 2.85 0.355 45.17 78.76 67.34 0.19 -2.04 -1.55 45.06 77.98 67.03 0.19 -2.00 -1.59 0.26 1.00 0.47 2.07 2.09 2.55 0.365 45.18 78.57 67.28 0.19 -2.05 -1.54 45.07 77.80 66.97 0.19 -2.00 -1.58 0.25 0.98 0.46 1.91 2.30 2.51

Table B.4 Error analysis of the finite element model in comparison with the analytical solution (the generalised Kirsch’s equations), for calculating the induced stresses along the radial distance r from the wall of a deviated borehole at � = 55.166°

163

r (m) Numerical Analytical Error calculation (%)

σrr σθθ σaa σrθ σra σaθ σrr σθθ σaa σrθ σra σaθ σrr (e) σθθ (e) σaa (e) σrθ (e) σra (e) σθa (e) 0.08 0.00 179.12 86.71 0.00 0.00 -1.85 0.00 179.24 86.70 0.00 0.00 -3.02 0.00 0.06 0.02 0.00 0.00 38.72 0.085 9.38 165.85 82.64 0.08 -0.10 -1.71 9.65 162.82 84.33 0.08 -0.24 -2.85 2.80 1.86 2.01 3.87 59.08 40.09 0.095 22.29 142.02 79.20 0.17 -1.11 -1.56 22.31 139.71 80.67 0.16 -0.61 -2.58 0.10 1.66 1.83 4.69 82.09 39.69 0.105 29.82 126.54 76.78 0.21 -1.47 -1.47 29.78 124.62 78.00 0.20 -0.88 -2.39 0.12 1.54 1.56 3.67 66.75 38.53 0.115 34.45 115.95 75.01 0.22 -1.68 -1.41 34.42 114.26 76.00 0.22 -1.08 -2.24 0.09 1.48 1.31 2.12 55.34 37.05 0.125 37.44 108.38 73.66 0.24 -1.82 -1.37 37.43 106.85 74.46 0.23 -1.24 -2.13 0.03 1.44 1.08 4.82 46.65 35.48 0.135 39.44 102.80 72.60 0.24 -1.91 -1.35 39.45 101.37 73.25 0.23 -1.36 -2.04 0.03 1.42 0.88 4.89 39.87 33.94 0.145 40.82 98.56 71.76 0.24 -1.97 -1.33 40.85 97.20 72.28 0.23 -1.46 -1.97 0.08 1.40 0.72 3.68 34.47 32.48 0.155 41.80 95.26 71.08 0.23 -2.01 -1.32 41.85 93.95 71.49 0.23 -1.54 -1.91 0.11 1.39 0.58 1.82 30.10 31.13 0.165 42.51 92.64 70.52 0.22 -2.03 -1.31 42.57 91.38 70.85 0.23 -1.61 -1.86 0.14 1.38 0.46 0.34 26.51 29.90 0.175 43.04 90.52 70.05 0.22 -2.05 -1.30 43.10 89.30 70.30 0.22 -1.66 -1.82 0.16 1.37 0.36 2.59 23.52 28.79 0.185 43.44 88.78 69.66 0.21 -2.07 -1.29 43.51 87.59 69.85 0.22 -1.71 -1.79 0.17 1.36 0.27 4.80 21.01 27.78 0.195 43.74 87.34 69.33 0.21 -2.08 -1.29 43.82 86.17 69.46 0.22 -1.75 -1.76 0.17 1.35 0.20 2.27 18.88 26.88 0.205 43.98 86.13 69.04 0.20 -2.09 -1.29 44.06 84.99 69.13 0.21 -1.78 -1.74 0.18 1.34 0.13 4.81 17.06 26.06 0.215 44.17 85.10 68.79 0.22 -2.09 -1.28 44.25 83.98 68.84 0.21 -1.81 -1.72 0.18 1.33 0.08 3.94 15.49 25.32 0.225 44.32 84.21 68.57 0.20 -2.10 -1.28 44.40 83.11 68.59 0.21 -1.84 -1.70 0.18 1.32 0.03 2.97 14.13 24.66 0.235 44.44 83.45 68.38 0.20 -2.10 -1.28 44.52 82.37 68.38 0.21 -1.86 -1.68 0.18 1.31 0.01 4.06 12.94 24.06 0.245 44.54 82.79 68.21 0.20 -2.10 -1.28 44.62 81.72 68.18 0.20 -1.88 -1.67 0.17 1.30 0.05 3.08 11.89 23.51 0.255 44.63 82.20 68.07 0.20 -2.10 -1.28 44.70 81.15 68.01 0.20 -1.89 -1.66 0.17 1.29 0.08 1.34 10.97 23.02 0.265 44.69 81.69 67.93 0.20 -2.10 -1.28 44.77 80.65 67.86 0.20 -1.91 -1.65 0.17 1.28 0.10 2.20 10.15 22.57 0.275 44.75 81.23 67.81 0.19 -2.11 -1.27 44.83 80.21 67.73 0.20 -1.92 -1.64 0.17 1.27 0.13 2.94 9.42 22.16 0.285 44.80 80.82 67.71 0.20 -2.11 -1.27 44.87 79.82 67.61 0.20 -1.94 -1.63 0.16 1.26 0.15 1.54 8.77 21.79 0.295 44.84 80.46 67.61 0.19 -2.11 -1.27 44.91 79.47 67.50 0.20 -1.95 -1.62 0.16 1.24 0.16 4.04 8.18 21.45 0.305 44.87 80.13 67.52 0.19 -2.11 -1.27 44.95 79.15 67.40 0.19 -1.96 -1.61 0.16 1.23 0.18 4.43 7.66 21.13 0.315 44.91 79.83 67.44 0.18 -2.11 -1.27 44.97 78.87 67.31 0.19 -1.97 -1.61 0.15 1.21 0.19 4.71 7.18 20.84 0.325 44.93 79.56 67.36 0.18 -2.11 -1.27 45.00 78.62 67.23 0.19 -1.97 -1.60 0.15 1.20 0.20 4.88 6.75 20.58 0.335 44.95 79.31 67.30 0.18 -2.11 -1.27 45.02 78.38 67.16 0.19 -1.98 -1.60 0.15 1.18 0.21 4.95 6.35 20.33 0.345 44.97 79.08 67.23 0.18 -2.11 -1.27 45.04 78.17 67.09 0.19 -1.99 -1.59 0.15 1.16 0.22 4.95 5.99 20.10 0.355 44.99 78.87 67.17 0.18 -2.11 -1.27 45.06 77.98 67.03 0.19 -2.00 -1.59 0.14 1.14 0.22 4.92 5.67 19.89 0.365 45.01 78.68 67.12 0.18 -2.11 -1.27 45.07 77.80 66.97 0.19 -2.00 -1.58 0.14 1.12 0.22 4.98 5.37 19.70

Table B.5 Error analysis of the finite element analysis based on the proposed boundary conditions in comparison with the analytical solution (the generalised Kirsch’s equations), for calculating the induced stresses along the radial distance r from the wall of a deviated borehole at � = 55.166°

APPENDIX C

True-triaxial data from the literature

APPENDIX C True-triaxial data from the literature

164

Tests No. σ₁ (MPa) σ₂ (MPa) σ₃ (MPa) I1(MPa) √J2(MPa) 1 310 0 0 310 178.98

2 397 20 20 437 217.66

3 417 51 20 488 220.80

4 413 92 20 525 209.23

5 453 165 20 638 220.40

6 460 206 20 686 220.87

7 465 233 20 718 222.57

8 449 40 40 529 236.14

9 446 40 40 526 234.40

10 486 80 40 606 246.76

11 499 113 40 652 246.65

12 530 193 40 763 250.69

13 547 274 40 861 253.75

14 535 315 40 890 248.01

15 473 60 60 593 238.45

16 517 87 60 664 256.41

17 537 102 60 699 264.11

18 530 113 60 703 257.42

19 576 164 60 800 272.89

20 550 197 60 807 252.81

21 553 275 60 888 247.17

22 557 345 60 962 249.39

23 528 80 80 688 258.65

24 572 126 80 778 271.75

25 577 150 80 807 269.02

26 647 208 80 935 297.38

27 591 225 80 896 263.34

28 677 283 80 1040 303.55

29 665 298 80 1043 295.65

30 650 378 80 1108 285.10

31 680 454 80 1214 303.03

Table C.1 True-triaxial data of Solnhofen Limestone, Mogi (2007)


165

Test NO. σ₁ (MPa) σ₂ (MPa) σ₃ (MPa) I1(MPa) √J2(MPa) 1 265 0 0 265 153.00

2 258 0 0 258 148.96

3 400 25 25 450 216.51

4 475 66 25 566 248.82

5 495 96 25 616 253.36

6 560 129 25 714 283.67

7 571 174 25 770 282.23

8 586 229 25 840 283.96

9 545 272 25 842 260.11

10 487 45 45 577 255.19

11 570 97 45 712 289.27

12 576 126 45 747 286.07

13 606 160 45 811 296.33

14 639 183 45 867 310.86

15 670 240 45 955 319.78

16 670 266 45 981 316.93

17 622 294 45 961 289.40

18 540 60 60 660 277.13

19 568 65 65 698 290.41

20 638 117 65 820 316.88

21 644 153 65 862 312.00

22 687 208 65 960 325.77

23 685 262 65 1012 316.79

24 746 318 65 1129 344.23

25 701 393 65 1159 318.05

26 620 85 85 790 308.88

27 684 128 85 897 334.11

28 719 153 85 957 348.07

29 744 233 85 1062 345.76

30 773 306 85 1164 351.25

31 818 376 85 1279 369.08

32 798 445 85 1328 356.51

33 682 105 105 892 333.13

34 778 167 105 1050 371.95

35 786 205 105 1096 367.72

36 805 268 105 1178 366.27

37 863 270 105 1238 398.63

38 824 334 105 1263 367.31

39 840 356 105 1301 373.60

40 822 415 105 1342 359.59

41 725 125 125 975 346.41

42 824 187 125 1136 386.91

43 860 239 125 1224 395.57

44 863 293 125 1281 386.82

45 897 362 125 1384 395.47

46 941 414 125 1480 413.74

47 918 463 125 1506 397.94

48 886 516 125 1527 380.55

49 883 253 145 1281 398.58

50 927 296 145 1368 414.83

51 923 324 145 1392 407.46

52 922 349 145 1416 402.84

53 1015 392 145 1552 448.34

54 1002 410 145 1557 438.77

Table C.2 True-triaxial data on Dunham Dolomite, Mogi (2007)


166

Test No. σ₁ (MPa) σ₂ (MPa) σ₃ (MPa) I1(MPa) √J2(MPa) 1 82 0 0 82 47.34

2 118 6 6 130 64.66

3 140 12.5 12.5 165 73.61

4 179 26 12.5 217.5 92.48

5 177 28 12.5 217.5 90.83

6 196 45 12.5 253.5 97.92

7 213 67 12.5 292.5 103.67

8 225 90 12.5 327.5 107.54

9 228 105 12.5 345.5 108.11

10 200 115 12.5 327.5 93.89

11 189 25 25 239 94.69

12 209 39 25 273 102.43

13 240 58 25 323 115.79

14 252 78 25 355 118.75

15 275 107 25 407 127.44

16 268 132 25 425 121.79

17 268 157 25 450 121.65

18 250 168 25 443 113.87

19 243 40 40 323 117.20

20 290 64 40 394 137.93

21 288 88 40 416 131.53

22 309 88 40 437 143.47

23 319 112 40 471 144.84

24 307 143 40 490 134.66

25 336 160 40 536 148.88

26 321 177 40 538 140.51

27 341 208 40 589 150.84

Table C.3 True-triaxial data on Yamaguchi Marble, Mogi (2007)


167


2 196 15 15 226 104.50

3 259 30 30 319 132.21

4 302 45 45 392 148.38

5 314 58 45 417 151.69

6 327 67 45 439 156.85

7 341 90 45 476 159.50

8 350 138 45 533 156.32

9 359 204 45 608 157.00

10 368 281 45 694 167.13

11 353 323 45 721 169.83

12 341 60 60 461 162.24

13 353 83 60 496 162.93

14 386 133 60 579 171.08

15 401 186 60 647 172.42

16 403 212 60 675 171.87

17 401 254 60 715 171.04

18 381 306 60 747 167.92

19 368 75 75 518 169.16

20 405 108 75 588 181.75

21 415 147 75 637 179.17

22 438 210 75 723 183.47

23 440 279 75 794 182.92

24 430 318 75 823 181.48

25 452 363 75 890 197.06

26 437 100 100 637 194.57

27 463 126 100 689 202.49

28 493 171 100 764 209.43

29 497 256 100 853 200.01

30 522 354 100 976 212.46

31 510 384 100 994 210.01

Table C.4 True-triaxial test data on Mizuho Trachyte (Mogi, 2007)


168


2 339 5 5 349 192.83

3 504 20 20 544 279.44

4 584.7 40 40 664.7 314.48

5 636 59 40 735 338.75

6 698 80 40 818 368.89

7 673 101 40 814 349.19

8 775 102 40 917 407.64

9 739 121 40 900 382.34

10 747 143 40 930 381.94

11 777 168 40 985 393.79

12 748 187 40 975 373.63

13 751 80 80 911 387.40

14 834 95 80 1009 431.06

15 810 108 80 998 413.62

16 836 117 80 1033 426.20

17 854 135 80 1069 431.87

18 893 147 80 1120 451.29

19 889 182 80 1151 440.59

20 930 183 80 1193 463.88

21 906 216 80 1202 442.88

22 973 218 80 1271 480.71

23 926 281 80 1287 441.99

24 956 284 80 1320 458.36

25 966 311 80 1357 459.60

26 962 140 140 1242 474.58

27 1098 205 140 1443 535.33

28 1144 259 140 1543 548.54

29 1161 331 140 1632 542.80

30 1168 424 140 1732 530.88

31 1107 200 200 1507 523.66

32 1168 235 200 1603 549.05

33 1244 251 200 1695 588.58

34 1305 298 200 1803 611.65

35 1352 343 200 1895 627.91

36 1329 401 200 1930 602.25

37 1358 473 200 2031 605.35

38 1364 537 200 2101 598.94

Table C.5 True-triaxial test data on Orikabe Monzonite (Mogi, 2007)


169


2 232 0 0 232 133.95

3 339 5 5 349 192.83

4 504 20 20 544 279.44

5 583 40 40 663 313.50

6 571 40 40 651 306.57

7 600 40 40 680 323.32

8 636 59 40 735 338.75

9 698 80 40 818 368.89

10 673 101 40 814 349.19

11 775 102 40 917 407.64

12 739 121 40 900 382.34

13 747 143 40 930 381.94

14 777 168 40 985 393.79

15 748 187 40 975 373.63

16 718 80 80 878 368.35

17 742 80 80 902 382.21

18 794 80 80 954 412.23

19 834 95 80 1009 431.06

20 810 108 80 998 413.62

21 836 117 80 1033 426.20

22 854 135 80 1069 431.87

23 893 147 80 1120 451.29

24 889 182 80 1151 440.59

25 930 183 80 1193 463.88

26 906 216 80 1202 442.88

27 973 218 80 1271 480.71

28 926 281 80 1287 441.99

29 956 284 80 1320 458.36

30 966 311 80 1357 459.60

31 943 140 140 1223 463.61

32 981 140 140 1261 485.55

33 1098 205 140 1443 535.33

34 1144 259 140 1543 548.54

35 1161 331 140 1632 542.80

36 1168 424 140 1732 530.88

37 1107 200 200 1507 523.66

38 1168 235 200 1603 549.05

39 1244 251 200 1695 588.58

40 1305 298 200 1803 611.65

41 1352 343 200 1895 627.91

42 1329 401 200 1930 602.25

43 1358 473 200 2031 605.35

44 1364 537 200 2101 598.94

Table C.6 True-triaxial test data on Inada Granite (Mogi, 2007)


170


2 349 16 16 381 192.26

3 364 20 20 404 198.61

4 381 20 20 421 208.42

5 470 67 20 557 247.36

6 516 124 20 660 261.56

7 538 186 20 744 264.51

8 552 40 40 632 295.60

9 577 75 40 692 300.44

10 632 112 40 784 323.02

11 669 126 40 835 341.05

12 653 206 40 899 317.05

13 626 278 40 944 294.72

14 671 70 70 811 346.99

15 735 101 70 906 375.31

16 735 152 70 957 362.59

17 808 193 70 1071 395.39

18 812 275 70 1157 383.18

19 801 313 70 1184 372.28

20 833 375 70 1278 384.05

21 806 100 100 1006 407.61

22 875 110 110 1095 441.67

23 881 130 130 1141 433.59

Table C.7 True-triaxial test data on Manazuru Andesite (Mogi, 2007)


171


2 346 79 0 425 181.31

3 291 149 0 440 145.51

4 347 197 0 544 174.03

5 267 229 0 496 144.44

6 410 30 30 470 219.39

7 479 60 30 569 251.02

8 599 100 30 729 310.29

9 652 200 30 882 321.48

10 571 249 30 850 272.13

11 637 298 30 965 304.19

12 702 60 60 822 370.66

13 750 88 60 898 390.54

14 766 103 60 929 395.78

15 745 155 60 960 371.11

16 816 199 60 1075 402.40

17 888 249 60 1197 433.90

18 828 299 60 1187 393.02

19 887 347 60 1294 419.90

20 954 399 60 1413 451.33

21 815 449 60 1324 377.56

22 868 100 100 1068 443.41

23 959 164 100 1223 478.54

24 1001 199 100 1300 494.10

25 945 248 100 1293 451.25

26 892 269 100 1261 417.12

27 1048 300 100 1448 499.70

28 1058 349 100 1507 497.07

29 1155 442 100 1697 538.26

30 1118 597 100 1815 509.05

31 1147 150 150 1447 575.62

32 1065 198 150 1413 514.98

33 1112 199 150 1461 541.82

34 1176 249 150 1575 565.95

35 1431 298 150 1879 700.78

36 1326 348 150 1824 629.64

37 1169 399 150 1718 531.23

38 1284 448 150 1882 587.89

39 1265 498 150 1913 570.47

40 1262 642 150 2054 557.23

Table C.8 True-triaxial test data on KTB Amphibolite (Chang and Haimson, 2000)


172


2 306 40 0 346 166.33

3 301 60 0 361 159.31

4 317 80 0 397 164.85

5 304 100 0 404 154.94

6 231 2 2 235 132.21

7 300 18 2 320 167.62

8 328 40 2 370 178.26

9 359 60 2 421 191.58

10 353 80 2 435 184.31

11 355 100 2 457 182.23

12 430 20 20 470 236.71

13 529 40 20 589 288.27

14 602 60 20 682 325.09

15 554 62 20 636 296.92

16 553 61 20 634 296.60

17 532 79 20 631 280.13

18 575 100 20 695 300.01

19 567 114 20 701 292.48

20 601 150 20 771 304.92

21 638 202 20 860 317.58

22 605 38 38 681 327.36

23 620 38 38 696 336.02

24 700 57 38 795 376.84

25 733 78 38 849 390.22

26 720 103 38 861 376.39

27 723 119 38 880 374.30

28 731 157 38 926 370.56

29 781 198 38 1017 391.05

30 747 60 60 867 396.64

31 811 90 60 961 425.19

32 821 114 60 995 424.63

33 860 180 60 1100 431.43

34 861 249 60 1170 418.70

35 889 77 77 1043 468.81

36 954 102 77 1133 499.28

37 992 142 77 1211 510.55

38 998 214 77 1289 496.93

39 1005 310 77 1392 482.79

40 1012 100 100 1212 526.54

41 1103 165 100 1368 561.26

42 1147 167 100 1414 586.10

43 1155 216 100 1471 578.53

44 1195 259 100 1554 591.66

45 1129 312 100 1541 543.33

Table C.9 True-triaxial test data on Westerly Granite (Haimson and Chang, 2000)


173

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 40 80 120 160 200

Line

ar c

orre

latio

n co

effic

ient

be

twee

n σ

2an

d σ

1

σ3 (MPa)

KTB AmphiboliteWesterly Granite

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 40 80 120 160 200

Line

ar c

orre

latio

n co

effic

ient

be

twee

n σ

2an

d σ

1

σ3 (MPa)

Dunham Dolomite

Solenhofen Limestone

Yamaguchi Marbel

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 40 80 120 160 200

Line

ar c

orre

latio

n co

effic

ient

be

twee

n σ

2an

d σ

1

σ3 (MPa)

Mizuho TrachyteManazuru AndesiteInada GraniteOrikabe Monzonite

Figure C.1 Linear correlation coefficient calculated by the means of Pearson linear correlation coefficient for the nine sets of true-triaxial data

APPENDIX D

)�-)� plots for the selected rock types from the literature

APPENDIX D ��-�� plots for the selected rock types from the literature

174

-200

0

200

400

600

800

1000

0 50 100 150 200 250

σ1(MPa)

σ2 (MPa)

KTB amphibolite, σ₃ = 0

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

200

400

600

800

1000

1200

0 100 200 300 400

σ1(MPa)

σ2 (MPa)

KTB amphibolite, σ₃ = 30 MPa

datapoints

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

Figure D.1 )�ÈÇ. )� Plots for KTB Amphibolite for different constant values of )�


175

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500

σ1(MPa)

σ2 (MPa)


datapoints

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

200

400

600

800

1000

1200

1400

1600

1800

0 200 400 600 800

σ1(MPa)

σ2 (MPa)


datapoints

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


176

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 200 400 600 800

σ1(MPa)

σ2 (MPa)


datapoints

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

100

200

300

400

500

600

700

0 50 100 150

σ1(MPa)

σ2 (MPa)

Westerly granite, σ₃ = 0 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

Figure D.2 )�ÈÇ. )� Plots for Westerly Granite for different constant values of )�


177

0

100

200

300

400

500

600

700

0 50 100 150

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

200

400

600

800

1000

1200

0 50 100 150 200 250

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


178

0

200

400

600

800

1000

1200

0 50 100 150 200 250

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

200

400

600

800

1000

1200

1400

0 50 100 150 200 250 300

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


179

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data poits

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


180

100

200

300

400

500

600

700

800

0 100 200 300

σ1(MPa)

σ2 (MPa)

Dunham Dolomite, σ₃ = 25 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

Figure D.3 )�ÈÇ. )� Plots for Dunham Dolomite for different constant values of )�


181

100

200

300

400

500

600

700

800

900

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

300

400

500

600

700

800

900

1000

0 100 200 300 400 500

σ1(MPa)

σ2 (MPa)

Dunham dolomite, σ₃ = 65 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


182

300

400

500

600

700

800

900

1000

1100

0 100 200 300 400 500

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

400

500

600

700

800

900

1000

1100

0 100 200 300 400 500

σ1(MPa)

σ2(MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


183

400

500

600

700

800

900

1000

1100

1200

1300

0 100 200 300 400 500 600

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

600

700

800

900

1000

1100

1200

200 300 400 500

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


184

250

300

350

400

450

500

550

600

650

0 50 100 150 200 250

σ1(MPa)

σ2 (MPa)

Solnhofen Limestone, σ₃ = 20 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplyfied Priest

Modified SP

250

300

350

400

450

500

550

600

650

700

750

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplyfied Priest

Modified SP

Figure D.4 )�ÈÇ. )� Plots for Solnhofen Limestone for different constant values of )�


185

250

350

450

550

650

750

850

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplyfied Priest

Modified SP

250

350

450

550

650

750

850

950

0 100 200 300 400 500

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplyfied Priest

Modified SP


186

0

50

100

150

200

250

300

350

0 50 100 150

σ1(MPa)

σ2 (MPa)

Yamaguchi marble, σ₃ = 12.5 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplyfied Priest

Modified SP

0

50

100

150

200

250

300

350

400

0 50 100 150 200

σ1(MPa)

σ2 (MPa)

Yamaguchi marble, σ₃ = 25 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplyfied Priest

Modified SP

Figure D.5 )�ÈÇ. )� Plots for Yamaguchi Marble for different constant values of )�


187

0

50

100

150

200

250

300

350

400

450

500

0 50 100 150 200 250

σ1(MPa)

σ2 (MPa)

Yamaguchi marble, σ₃ = 40 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplyfied Priest

Modified SP

0

100

200

300

400

500

600

0 100 200 300 400

σ1(MPa)

σ2 (MPa)

Mizuho Trachyte, σ₃ = 45 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

Figure D.6 )�ÈÇ. )� Plots for Mizuho Trachyte for different constant values of )�


188

100

150

200

250

300

350

400

450

500

550

600

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

100

200

300

400

500

600

700

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


189

200

250

300

350

400

450

500

550

600

650

700

0 100 200 300 400 500

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

100

200

300

400

500

600

700

800

900

0 50 100 150 200

σ1(MPa)

σ2 (MPa)

Manazuru Andesite, σ₃ = 20 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

Figure D.7 )�ÈÇ. )� Plots for Manazuru Andesite for different constant values of )�


190

0

200

400

600

800

1000

1200

0 100 200 300

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

0

200

400

600

800

1000

1200

1400

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


191

0

100

200

300

400

500

600

700

800

900

0 50 100 150

σ1(MPa)

σ2 (MPa)

Inada Granite, σ₃ = 20 MPa

data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

100

200

300

400

500

600

700

800

900

1000

1100

0 50 100 150 200

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

Figure D.8 )�ÈÇ. )� Plots for Inada Granite for different constant values of )�


192

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


193

400

600

800

1000

1200

1400

1600

1800

100 150 200 250 300 350 400

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP

600

800

1000

1200

1400

1600

1800

2000

150 250 350 450 550

σ1(MPa)

σ2 (MPa)


data points

Hoek-Brown

Pan-Hudson

Zhang-Zhu

Generalised Priest

Simplified Priest

Modified SP


194

0

100

200

300

400

500

600

700

800

900

0 50 100 150 200

σ1(MPa)

σ2 (MPa)

Orikabe Monzonite, σ₃ = 40 MPa

data points

Modified SP

Zhang-Zhu

Hoek-Brown

Pan-Hudson

Generalised Priest

Simplyfied Preist

200

300

400

500

600

700

800

900

1000

1100

1200

0 100 200 300 400

σ1(MPa)

σ2 (MPa)


data points

Modified SP

Zhang-Zhu

Hoek-Brown

Pan-Hudson

Generalised Priest

Simplyfied Preist

Figure D.9 )�ÈÇ. )� Plots for Orikabe Monzonite for different constant values of )�


195

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500

σ1(MPa)

σ2 (MPa)


data points

Modified SP

Zhang-Zhu

Hoek-Brown

Pan-Hudson

Generalised Priest

Simplyfied Preist

600

800

1000

1200

1400

1600

1800

0 100 200 300 400 500 600

σ1(MPa)

σ2 (MPa)


data points

Modified SP

Zhang-Zhu

Hoek-Brown

Pan-Hudson

Generalised Priest

Simplyfied Preist

APPENDIX E

Error analysis diagrams

APPENDIX E Error analysis diagrams (PDF)

196

-500 -400 -300 -200 -100 0 100 200 300 400 5000

1

2

3

4

5

6

7x 10

-3

X = σ1(model)

- σ1(observed)

pro

bab

ility

den

sity

fun

ctio

n f

(X)

PH (µ=-251.342,σ=71.7861) HB (µ=-47.5298,σ=93.4314) ZZ (µ=-25.5263,σ=57.7988) GP (µ=45.2787,σ=93.049) SP (µ=-54.1183,σ=60.1017)MSP (µ=-2.47782,σ=59.7621)

-500 -400 -300 -200 -100 0 100 200 300 400 5000

0.002

0.004

0.006

0.008

0.01

0.012

X = σ1(model)

- σ1(observed)

pro

bab

ility

den

sity

fun

ctio

n f

(X)


Figure E.2 Normal distribution of failure prediction accuracy of selected failure criteria for Inada Granite

Figure E.1 Normal distribution of failure prediction accuracy of selected failure criteria for Orikabe Monzonite


197

-500 -400 -300 -200 -100 0 100 200 300 400 5000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

X = σ1(model)

- σ1(observed)

pro

bab

ility

den

sity

fun

ctio

n f

(X)

PH (µ=-170.007,σ=99.2906) HB (µ=-100.039,σ=62.4637) ZZ (µ=16.8001,σ=56.9715) GP (µ=106.245,σ=135.468) SP (µ=-16.1147,σ=42.2597)MSP (µ=8.93291,σ=42.6999)

-500 -400 -300 -200 -100 0 100 200 300 400 5000

0.005

0.01

0.015

0.02

0.025

X = σ1(model)- σ1(observed)

pro

bab

ility

den

sity

fun

ctio

n f

(X)

PH (µ=-20.9175,σ=55.7117) HB (µ=-43.9824,σ=26.3057) ZZ (µ=4.57147,σ=20.7621) GP (µ=65.3997,σ=69.0968) SP (µ=-10.3116,σ=16.0249)MSP (µ=-10.8578,σ=21.2673)

Figure E.3 Normal distribution of failure prediction accuracy of selected failure criteria for Manazuru Andesite

Figure E.4 Normal distribution of failure prediction accuracy of selected failure criteria for Mizuho Trachyte


198

-500 -400 -300 -200 -100 0 100 200 300 400 5000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

X = σ1(model)

- σ1(observed)

pro

bab

ility

den

sity

fu

nct

ion

f (

X)


-500 -400 -300 -200 -100 0 100 200 300 400 5000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

X = σ1(model)

- σ1(observed)

pro

bab

ility

den

sity

fu

nct

ion

f (

X)


Figure E.5 Normal distribution of failure prediction accuracy of selected failure criteria for Yamaguchi Marble

Figure E.6 Normal distribution of failure prediction accuracy of selected failure criteria for Solnhofen Limestone


199

-500 -400 -300 -200 -100 0 100 200 300 400 5000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

X = σ1(model)

- σ1(observed)

pro

bab

ility

den

sity

fun

ctio

n f

(X)


-500 -400 -300 -200 -100 0 100 200 300 400 5000

1

2

3

4

5

6

7

8x 10

-3

X = σ1(model)- σ1(observed)

pro

bab

ility

den

sity

fu

nct

ion

f (X

)


Figure E.7 Normal distribution of failure prediction accuracy of selected failure criteria for Dunham Dolomite

Figure E.8 Normal distribution of failure prediction accuracy of selected failure criteria for Westerly Granite


200

-500 -400 -300 -200 -100 0 100 200 300 400 5000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

-3

X = σ1(model)

- σ1(observed)

pro

bab

ility

den

sity

fun

ctio

n f

(X

)


Figure E.9 Normal distribution of failure prediction accuracy of selected failure criteria for KTB Amphibolite

APPENDIX F

MATLAB programs for plotting three dimensional failure

surfaces in the principal stress space

201

Hoek-Brown Criterion

%===========Hoek Brown Criterion=================== ======================== % Input parameters: % I1= First invariant of stress tensor. % mb= Parameter m for the Hoek-Brown Criterion. % sigmac= Uniaxial strength of rock material. % parameter S for the Hoek-Brown Criterion. % Output: % Failure surface for the Hoek-Brown criterion in t hree-dimensional stress space. %================================================== ======================== I1=linspace(0,3500,50); mb=20; sigmac=100; S=1; %%====Calculation of the Hoek-Brown radius on the d eviatoric plane========= %---[-pi/6,pi/6]----------------------------------- ------------------------ theta=(-pi/6:.01:pi/6); lamda1=2*mb*sin(theta+pi/3)./sqrt(3); A=4*(cos(theta)).^2; for j=1:length(I1) for i=1:length(A) r1(i,j)=(sigmac./(2*A(i))).*(-lamda1(i)+sqrt(lamda1(i).^2+4*A(i).*(mb*I1(j)/(3*si gmac))+S)); end end %---[pi/6,pi/2]------------------------------------ ------------------------ theta1=(-(pi/6):0.01:pi/6); lamda2=2*mb*sin(-theta+pi/3)./sqrt(3); A=4*(cos(theta)).^2; for j=1:length(I1) for i=1:length(A) r2(i,j)=(sigmac./(2*A(i))).*(-lamda2(i)+sqrt(lamda2(i).^2+4*A(i).*(mb*I1(j)/(3*si gmac))+S)); end end %---[pi/2,2*pi/3]---------------------------------- --------------------- theta2=(-(pi/6):0.01:pi/6); lamda1=2*mb*sin(theta+pi/3)./sqrt(3); for j=1:length(I1) for i=1:length(A) r3(i,j)=(sigmac./(2*A(i))).*(-lamda1(i)+sqrt(lamda1(i).^2+4*A(i).*(mb*I1(j)/(3*si gmac))+S)); end end %---[5*pi/6,7*pi/6]-------------------------------- ------------------------ theta3=((-pi/6):0.01:pi/6); lamda2=2*mb*sin(-theta+pi/3)./sqrt(3); for j=1:length(I1) for i=1:length(A) r4(i,j)=(sigmac./(2*A(i))).*(-lamda2(i)+sqrt(lamda2(i).^2+4*A(i).*(mb*I1(j)/(3*si gmac))+S)); end end %---[7pi/6,3pi/2]---------------------------------- ------------------------

202

theta4=((-pi/6):0.01:pi/6); lamda1=2*mb*sin(theta+pi/3)./sqrt(3); for j=1:length(I1) for i=1:length(A) r5(i,j)=(sigmac./(2*A(i))).*(-lamda1(i)+sqrt(lamda1(i).^2+4*A(i).*(mb*I1(j)/(3*si gmac))+S)); end end %---[3pi/2,-pi/6]---------------------------------- ------------------------ theta5=(-pi/6:0.01:pi/6); lamda2=2*mb*sin(pi/3-theta)./sqrt(3); for j=1:length(I1) for i=1:length(A) r6(i,j)=(sigmac./(2*A(i))).*(-lamda2(i)+sqrt(lamda2(i).^2+4*A(i).*(mb*I1(j)/(3*si gmac))+S)); end end %-------------------------------------------------- ------------------------ figure(1) % 2-D Hoek-Brown trace on the deviatoric plane. for h=1:length(I1) [X,Y] = pol2cart(theta,r1(:,h)'); plot(X,Y) hold on [X1,Y1] = pol2cart(theta1+pi/3,r2(:,h)'); plot(X1,Y1) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,r3(:,h)'); plot(X2,Y2) hold on [X3,Y3] = pol2cart(theta3+pi,r4(:,h)'); plot(X3,Y3) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,r5(:,h)'); plot(X4,Y4) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,r6(:,h)'); plot(X5,Y5) end xlabel( '\sigma_2d (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1d (Mpa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); figure(2) % 3-D Hoek-Brown surface in stress space. for h=1:length(I1) [X,Y] = pol2cart(theta,sqrt(2).*r1(:,h)'); I16=(((((2*A.*r1(:,h)'/sigmac)+lamda1).^2-lamda 1.^2)./(4*A))-S)*3*sigmac/mb; z1=I16; A1=R*[X;Y;z1/sqrt(3)]; plot3(A1(1,:),A1(2,:),A1(3,:), 'r' ) hold on [X1,Y1] = pol2cart(theta1+pi/3,sqrt(2).*r2(:,h) '); I16=(((((2*A.*r2(:,h)'/sigmac)+lamda2).^2-lamda 2.^2)./(4*A))-S)*3*sigmac/mb; z2=I16;

203

B=R*[X1;Y1;z2/sqrt(3)]; plot3(B(1,:),B(2,:),B(3,:), 'r' ) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,sqrt(2).*r3(:, h)'); I16=(((((2*A.*r3(:,h)'/sigmac)+lamda1).^2-lamda 1.^2)./(4*A))-S)*3*sigmac/mb; z3=I16; C=R*[X2;Y2;z3/sqrt(3)]; plot3(C(1,:),C(2,:),C(3,:), 'r' ) hold on [X3,Y3] = pol2cart(theta3+pi,sqrt(2).*r4(:,h)') ; I16=(((((2*A.*r4(:,h)'/sigmac)+lamda2).^2-lamda 2.^2)./(4*A))-S)*3*sigmac/mb; z4=I16; D=R*[X3;Y3;z4/sqrt(3)]; plot3(D(1,:),D(2,:),D(3,:), 'r' ) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,sqrt(2).*r5(:, h)'); I16=(((((2*A.*r5(:,h)'/sigmac)+lamda1).^2-lamda 1.^2)./(4*A))-S)*3*sigmac/mb; z5=I16; E=R*[X4;Y4;z5/sqrt(3)]; plot3(E(1,:),E(2,:),E(3,:), 'r' ) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,sqrt(2).*r6(:, h)'); I16=(((((2*A.*r6(:,h)'/sigmac)+lamda2).^2-lamda 2.^2)./(4*A))-S)*3*sigmac/mb; z6=I16; F=R*[X5;Y5;z6/sqrt(3)]; plot3(F(1,:),F(2,:),F(3,:), 'r' ) end % Moving axes to the origin of coordinate system. plot3(get(gca, 'XLim' ),[0 0],[0 0], 'k' ); plot3([0 0],[0 0],get(gca, 'ZLim' ), 'k' ); plot3([0 0],get(gca, 'YLim' ),[0 0], 'k' ); % REMOVE TICKS set(gca, 'Xtick' ,[]); set(gca, 'Ytick' ,[]); set(gca, 'Ztick' ,[]); % GET OFFSETS Xoff=diff(get(gca, 'XLim' ))./30; Yoff=diff(get(gca, 'YLim' ))./30; Zoff=diff(get(gca, 'ZLim' ))./30; xlabel( '\sigma_2 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) zlabel( '\sigma_3 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14)

204

Pan-Hudson Criterion

%===========Pan-Hudson Criterion=================== ======================== % Input parameters: % I1= First invariant of stress tensor. % mb= Parameter m for the Hoek-Brown Criterion. % sigmac= Uniaxial strength of rock material. % parameter S for the Hoek-Brown Criterion. % Output: % Failure surface for the Hoek-Brown criterion in t hree-dimensional stress space. %================================================== ======================== clear all close all I1=linspace(0,3500,50); mb=20; sigmac=100; S=1; e=0.00; %%====Calculation of the Pan-Hudson radius on the d eviatoric plane========= %===[-pi/6,pi/6]================================ theta=(-pi/6:.01:pi/6); A=3*ones(1,length(theta)); lamda=(sqrt(3)/2)*mb*ones(1,length(theta)); for j=1:length(I1) for i=1:length(lamda) r1(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===[pi/6,pi/2]================================ theta1=(-(pi/6)+e:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r2(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===============[pi/2,2pi/3]======================= ========== theta2=(-(pi/6)-e:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r3(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===============[5pi/6,7pi/6]====================== ============= theta3=((-pi/6)-e:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r4(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %=================[7pi/6,3pi/2]==================== theta4=((-pi/6)-e:0.01:pi/6);

205

for j=1:length(I1) for i=1:length(lamda) r5(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===============[3pi/2,-pi/6]====================== =============== theta5=(-pi/6:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r6(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %-------------------------------------------------- ------------------------ figure(1) % 2-D Pan-Hudson trace on the deviatoric plane. for h=1:length(I1) [X,Y] = pol2cart(theta,r1(:,h)'); plot(X,Y) hold on [X1,Y1] = pol2cart(theta1+pi/3,r2(:,h)'); plot(X1,Y1) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,r3(:,h)'); plot(X2,Y2) hold on [X3,Y3] = pol2cart(theta3+pi,r4(:,h)'); plot(X3,Y3) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,r5(:,h)'); plot(X4,Y4) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,r6(:,h)'); plot(X5,Y5) end xlabel( '\sigma_2d (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1d (Mpa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); figure(2) % 3-D Pan-Hudson surface in stress space. for h=1:length(I1) [X,Y] = pol2cart(theta,sqrt(2).*r1(:,h)'); I16=(((((6*r1(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z1=I16; A1=R*[X;Y;z1/sqrt(3)]; plot3(A1(1,:),A1(2,:),A1(3,:), 'r' ) hold on [X1,Y1] = pol2cart(theta1+pi/3,sqrt(2).*r2(:,h) '); I16=(((((6*r2(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z2=I16; B=R*[X1;Y1;z2/sqrt(3)]; plot3(B(1,:),B(2,:),B(3,:), 'r' ) hold on

206

[X2,Y2] = pol2cart(theta2+2*pi/3,sqrt(2).*r3(:, h)'); I16=(((((6*r3(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z3=I16; C=R*[X2;Y2;z3/sqrt(3)]; plot3(C(1,:),C(2,:),C(3,:), 'r' ) hold on [X3,Y3] = pol2cart(theta3+pi,sqrt(2).*r4(:,h)') ; I16=(((((6*r4(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z4=I16; D=R*[X3;Y3;z4/sqrt(3)]; plot3(D(1,:),D(2,:),D(3,:), 'r' ) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,sqrt(2).*r5(:, h)'); I16=(((((6*r5(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z5=I16; E=R*[X4;Y4;z5/sqrt(3)]; plot3(E(1,:),E(2,:),E(3,:), 'r' ) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,sqrt(2).*r6(:, h)'); I16=(((((6*r6(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z6=I16; F=R*[X5;Y5;z6/sqrt(3)]; plot3(F(1,:),F(2,:),F(3,:), 'r' ) end % Moving axes to the origin of coordinate system. plot3(get(gca, 'XLim' ),[0 0],[0 0], 'k' ); plot3([0 0],[0 0],get(gca, 'ZLim' ), 'k' ); plot3([0 0],get(gca, 'YLim' ),[0 0], 'k' ); X=get(gca, 'Xtick' ); Y=get(gca, 'Ytick' ); Z=get(gca, 'Ztick' ); XL=get(gca, 'XtickLabel' ); YL=get(gca, 'YtickLabel' ); ZL=get(gca, 'ZtickLabel' ); % REMOVE TICKS set(gca, 'Xtick' ,[]); set(gca, 'Ytick' ,[]); set(gca, 'Ztick' ,[]); % GET OFFSETS Xoff=diff(get(gca, 'XLim' ))./30; Yoff=diff(get(gca, 'YLim' ))./30; Zoff=diff(get(gca, 'ZLim' ))./30; xlabel( '\sigma_2 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) zlabel( '\sigma_3 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14)

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Zhang-Zhu Criterion

%===========Zhang-Zhu Criterion==================== ======================= % Input parameters: % I1= First invariant of stress tensor. % mb= Parameter m for the Hoek-Brown Criterion. % sigmac= Uniaxial strength of rock material. % parameter S for the Hoek-Brown Criterion. % Output: % Failure surface for the Hoek-Brown criterion in t hree-dimensional stress space. %================================================== ======================== I1=linspace(0,3500,50); mb=20; sigmac=100; S=1; e=0.00; %%====Calculation of the Zhang-Zhu radius on the de viatoric plane========= %===[-pi/6,pi/6]================================ theta=(-pi/6:.01:pi/6); lamda1=(sqrt(3)/2)*mb+(sqrt(3)/3)*mb*sin(theta); %A=4*(cos(theta)).^2; for j=1:length(I1) for i=1:length(lamda1) r1(i,j)=(sigmac./6).*(-lamda1(i)+sqrt(lamda1(i).^2+12.*(mb*I1(j)/(3*sigmac ))+S)); end end %===[pi/6,pi/2]================================ theta1=(-(pi/6)+e:0.01:pi/6); lamda2=(sqrt(3)/2)*mb-(sqrt(3)/3)*mb*sin(theta); %A=4*(cos(theta)).^2; for j=1:length(I1) for i=1:length(lamda2) r2(i,j)=(sigmac./6).*(-lamda2(i)+sqrt(lamda2(i).^2+12.*(mb*I1(j)/(3*sigmac ))+S)); end end %===============[pi/2,2pi/3]======================= ========== theta2=(-(pi/6)-e:0.01:pi/6); lamda1=(sqrt(3)/2)*mb+(sqrt(3)/3)*mb*sin(theta); for j=1:length(I1) for i=1:length(lamda1) r3(i,j)=(sigmac./6).*(-lamda1(i)+sqrt(lamda1(i).^2+12.*(mb*I1(j)/(3*sigmac ))+S)); end end %===============[5pi/6,7pi/6]====================== ============= theta3=((-pi/6)-e:0.01:pi/6); lamda2=(sqrt(3)/2)*mb-(sqrt(3)/3)*mb*sin(theta); for j=1:length(I1) for i=1:length(lamda2) r4(i,j)=(sigmac./6).*(-lamda2(i)+sqrt(lamda2(i).^2+12.*(mb*I1(j)/(3*sigmac ))+S)); end end

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%=================[7pi/6,3pi/2]==================== theta4=((-pi/6)-e:0.01:pi/6); lamda1=(sqrt(3)/2)*mb+(sqrt(3)/3)*mb*sin(theta); for j=1:length(I1) for i=1:length(lamda1) r5(i,j)=(sigmac./6).*(-lamda1(i)+sqrt(lamda1(i).^2+12.*(mb*I1(j)/(3*sigmac ))+S)); end end %===============[3pi/2,-pi/6]====================== =============== theta5=(-pi/6:0.01:pi/6); lamda2=(sqrt(3)/2)*mb-(sqrt(3)/3)*mb*sin(theta); for j=1:length(I1) for i=1:length(lamda2) r6(i,j)=(sigmac./6).*(-lamda2(i)+sqrt(lamda2(i).^2+12.*(mb*I1(j)/(3*sigmac ))+S)); end end %-------------------------------------------------- ------------------------ figure(1) % 2-D Hoek-Brown trace on the deviatoric plane. for h=1:length(I1) [X,Y] = pol2cart(theta,r1(:,h)'); plot(X,Y) hold on [X1,Y1] = pol2cart(theta1+pi/3,r2(:,h)'); plot(X1,Y1) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,r3(:,h)'); plot(X2,Y2) hold on [X3,Y3] = pol2cart(theta3+pi,r4(:,h)'); plot(X3,Y3) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,r5(:,h)'); plot(X4,Y4) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,r6(:,h)'); plot(X5,Y5) end xlabel( '\sigma_2d (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1d (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); figure(2) % 3-D Hoek-Brown surface in stress space. for h=1:length(I1) [X,Y] = pol2cart(theta,sqrt(2).*r1(:,h)'); I16=(((((6*r1(:,h)'/sigmac)+lamda1).^2-lamda1.^ 2)./(12))-S)*3*sigmac/mb; z1=I16; A1=R*[X;Y;z1/sqrt(3)]; plot3(A1(1,:),A1(2,:),A1(3,:), 'r' ) hold on [X1,Y1] = pol2cart(theta1+pi/3,sqrt(2).*r2(:,h) ');

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I16=(((((6*r2(:,h)'/sigmac)+lamda2).^2-lamda2.^ 2)./(12))-S)*3*sigmac/mb; z2=I16; B=R*[X1;Y1;z2/sqrt(3)]; plot3(B(1,:),B(2,:),B(3,:), 'r' ) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,sqrt(2).*r3(:, h)'); I16=(((((6*r3(:,h)'/sigmac)+lamda1).^2-lamda1.^ 2)./(12))-S)*3*sigmac/mb; z3=I16; C=R*[X2;Y2;z3/sqrt(3)]; plot3(C(1,:),C(2,:),C(3,:), 'r' ) hold on [X3,Y3] = pol2cart(theta3+pi,sqrt(2).*r4(:,h)') ; I16=(((((6*r4(:,h)'/sigmac)+lamda2).^2-lamda2.^ 2)./(12))-S)*3*sigmac/mb; z4=I16; D=R*[X3;Y3;z4/sqrt(3)]; plot3(D(1,:),D(2,:),D(3,:), 'r' ) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,sqrt(2).*r5(:, h)'); I16=(((((6*r5(:,h)'/sigmac)+lamda1).^2-lamda1.^ 2)./(12))-S)*3*sigmac/mb; z5=I16; E=R*[X4;Y4;z5/sqrt(3)]; plot3(E(1,:),E(2,:),E(3,:), 'r' ) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,sqrt(2).*r6(:, h)'); I16=(((((6*r6(:,h)'/sigmac)+lamda2).^2-lamda2.^ 2)./(12))-S)*3*sigmac/mb; z6=I16; F=R*[X5;Y5;z6/sqrt(3)]; plot3(F(1,:),F(2,:),F(3,:), 'r' ) end % Moving axes to the origin of coordinate system. plot3(get(gca, 'XLim' ),[0 0],[0 0], 'k' ); plot3([0 0],[0 0],get(gca, 'ZLim' ), 'k' ); plot3([0 0],get(gca, 'YLim' ),[0 0], 'k' ); X=get(gca, 'Xtick' ); Y=get(gca, 'Ytick' ); Z=get(gca, 'Ztick' ); XL=get(gca, 'XtickLabel' ); YL=get(gca, 'YtickLabel' ); ZL=get(gca, 'ZtickLabel' ); % REMOVE TICKS set(gca, 'Xtick' ,[]); set(gca, 'Ytick' ,[]); set(gca, 'Ztick' ,[]); % GET OFFSETS Xoff=diff(get(gca, 'XLim' ))./30; Yoff=diff(get(gca, 'YLim' ))./30; Zoff=diff(get(gca, 'ZLim' ))./30; xlabel( '\sigma_2 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14)

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zlabel( '\sigma_3 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14)

Simplified Priest Criterion

%===========Simplified-Priest Criterion========================================== = % Input parameters: % I1= First invariant of stress tensor. % mb= Parameter m for the Hoek-Brown Criterion. % sigmac= Uniaxial strength of rock material. % parameter S for the Hoek-Brown Criterion. % Output: % Failure surface for the Hoek-Brown criterion in t hree-dimensional stress space. %================================================== ======================== I1=linspace(0,3500,50); mb=20; sigmac=80; S=1; w=0.299289347; e=0.00; %%====Calculation of the Simplified Priest radius o n the deviatoric plane========= %===[-pi/6,pi/6]================================ theta=(-pi/6:.01:pi/6); lamda1=2*mb.*((sqrt(3)/3)*sin(theta+pi/3)-w*sin(the ta+pi/6)); A1=9*w^2*(sqrt(3)*sin(theta)+cos(theta)).^2-6*w*(3+4*sqrt(3)*sin(theta).*cos(theta))+3*(sin(the ta)+sqrt(3)*cos(theta)).^2; for j=1:length(I1) for i=1:length(lamda1) r1(i,j)=(sigmac./(2*A1(i))).*(-lamda1(i)+sqrt(lamda1(i).^2+4*A1(i).*(mb*I1(j)/(3*s igmac))+S)); end end %===[pi/6,pi/2]================================ theta1=(-(pi/6)+e:0.01:pi/6); lamda2=2*mb.*(-(sqrt(3)/3)*sin(theta-pi/3)+w*sin(th eta-pi/6)); A2=9*w^2*(sqrt(3)*sin(theta)-cos(theta)).^2-6*w*(3-4*sqrt(3)*sin(theta).*cos(theta))+3*(sin(theta)-sqr t(3)*cos(theta)).^2;; for j=1:length(I1) for i=1:length(lamda2) r2(i,j)=(sigmac./(2*A2(i))).*(-lamda2(i)+sqrt(lamda2(i).^2+4*A2(i).*(mb*I1(j)/(3*s igmac))+S)); end end %===============[pi/2,2pi/3]======================= ========== theta2=(-(pi/6)-e:0.01:pi/6); lamda1=2*mb.*((sqrt(3)/3)*sin(theta+pi/3)-w*sin(the ta+pi/6)); for j=1:length(I1)

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for i=1:length(lamda1) r3(i,j)=(sigmac./(2*A1(i))).*(-lamda1(i)+sqrt(lamda1(i).^2+4*A1(i).*(mb*I1(j)/(3*s igmac))+S)); end end %===============[5pi/6,7pi/6]====================== ============= theta3=((-pi/6)-e:0.01:pi/6); lamda2=2*mb.*(-(sqrt(3)/3)*sin(theta-pi/3)+w*sin(th eta-pi/6)); for j=1:length(I1) for i=1:length(lamda2) r4(i,j)=(sigmac./(2*A2(i))).*(-lamda2(i)+sqrt(lamda2(i).^2+4*A2(i).*(mb*I1(j)/(3*s igmac))+S)); end end %=================[7pi/6,3pi/2]==================== theta4=((-pi/6)-e:0.01:pi/6); lamda1=2*mb.*((sqrt(3)/3)*sin(theta+pi/3)-w*sin(the ta+pi/6)); for j=1:length(I1) for i=1:length(lamda1) r5(i,j)=(sigmac./(2*A1(i))).*(-lamda1(i)+sqrt(lamda1(i).^2+4*A1(i).*(mb*I1(j)/(3*s igmac))+S)); end end %===============[3pi/2,-pi/6]====================== =============== theta5=(-pi/6:0.01:pi/6); lamda2=2*mb.*(-(sqrt(3)/3)*sin(theta-pi/3)+w*sin(th eta-pi/6)); for j=1:length(I1) for i=1:length(lamda2) r6(i,j)=(sigmac./(2*A2(i))).*(-lamda2(i)+sqrt(lamda2(i).^2+4*A2(i).*(mb*I1(j)/(3*s igmac))+S)); end end %-------------------------------------------------- ------------------------ figure(1) % 2-D Simplified Priest trace on the deviatoric pla ne. for h=1:length(I1) [X,Y] = pol2cart(theta,r1(:,h)'); plot(X,Y) hold on [X1,Y1] = pol2cart(theta1+pi/3,r2(:,h)'); plot(X1,Y1) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,r3(:,h)'); plot(X2,Y2) hold on [X3,Y3] = pol2cart(theta3+pi,r4(:,h)'); plot(X3,Y3) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,r5(:,h)'); plot(X4,Y4) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,r6(:,h)'); plot(X5,Y5) end xlabel( '\sigma_2d (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1d (Mpa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)];

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R=inv(R1); R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); figure(2) % 3-D Simplified-Priest surface in stress space. for h=1:length(I1) [X,Y] = pol2cart(theta,sqrt(2).*r1(:,h)'); I16=((3*sigmac).*(((((r1(:,h)'./sigmac).*(2*A1) +lamda1).^2-lamda1.^2)./(4.*A1))-S))./mb; z1=I16; T1=R*[X;Y;z1/sqrt(3)]; plot3(T1(1,:),T1(2,:),T1(3,:), 'r' ) hold on [X1,Y1] = pol2cart(theta1+pi/3,sqrt(2).*r2(:,h) '); I16=((3*sigmac).*(((((r2(:,h)'./sigmac).*(2*A2) +lamda2).^2-lamda2.^2)./(4.*A2))-S))./mb; z2=I16; B=R*[X1;Y1;z2/sqrt(3)]; plot3(B(1,:),B(2,:),B(3,:), 'r' ) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,sqrt(2).*r3(:, h)'); I16=((3*sigmac).*(((((r3(:,h)'./sigmac).*(2*A1) +lamda1).^2-lamda1.^2)./(4.*A1))-S))./mb; z3=I16; C=R*[X2;Y2;z3/sqrt(3)]; plot3(C(1,:),C(2,:),C(3,:), 'r' ) hold on [X3,Y3] = pol2cart(theta3+pi,sqrt(2).*r4(:,h)') ; I16=((3*sigmac).*(((((r4(:,h)'./sigmac).*(2*A2) +lamda2).^2-lamda2.^2)./(4.*A2))-S))./mb; z4=I16; D=R*[X3;Y3;z4/sqrt(3)]; plot3(D(1,:),D(2,:),D(3,:), 'r' ) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,sqrt(2).*r5(:, h)'); I16=((3*sigmac).*(((((r5(:,h)'./sigmac).*(2*A1) +lamda1).^2-lamda1.^2)./(4.*A1))-S))./mb; z5=I16; E=R*[X4;Y4;z5/sqrt(3)]; plot3(E(1,:),E(2,:),E(3,:), 'r' ) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,sqrt(2).*r6(:, h)'); I16=((3*sigmac).*(((((r6(:,h)'./sigmac).*(2*A2) +lamda2).^2-lamda2.^2)./(4.*A2))-S))./mb; z6=I16; F=R*[X5;Y5;z6/sqrt(3)]; plot3(F(1,:),F(2,:),F(3,:), 'r' ) end % Moving axes to the origin of coordinate system. plot3(get(gca, 'XLim' ),[0 0],[0 0], 'k' ); plot3([0 0],[0 0],get(gca, 'ZLim' ), 'k' ); plot3([0 0],get(gca, 'YLim' ),[0 0], 'k' ); X=get(gca, 'Xtick' ); Y=get(gca, 'Ytick' ); Z=get(gca, 'Ztick' ); XL=get(gca, 'XtickLabel' ); YL=get(gca, 'YtickLabel' );

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ZL=get(gca, 'ZtickLabel' ); % REMOVE TICKS set(gca, 'Xtick' ,[]); set(gca, 'Ytick' ,[]); set(gca, 'Ztick' ,[]); % GET OFFSETS Xoff=diff(get(gca, 'XLim' ))./30; Yoff=diff(get(gca, 'YLim' ))./30; Zoff=diff(get(gca, 'ZLim' ))./30; xlabel( '\sigma_2 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) zlabel( '\sigma_3 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14)

Generalised Priest Criterion

%===========Generalised-Priest Criterion========================================== = % Input parameters: % I1= First invariant of stress tensor. % mb= Parameter m for the Hoek-Brown Criterion. % sigmac= Uniaxial strength of rock material. % parameter S for the Hoek-Brown Criterion. % Output: % Failure surface for the Hoek-Brown criterion in t hree-dimensional stress space. %================================================== ======================== clear all close all I1=linspace(0,3500,50); mb=20; sigmac=100; S=1; e=0.00; %%====Calculation of the Generalised Priest radius on the deviatoric plane========= %===[-pi/6,pi/6]================================ theta=(-pi/6:.01:pi/6); A=3*ones(1,length(theta)); lamda=(sqrt(3)/3)*mb*ones(1,length(theta)); for j=1:length(I1) for i=1:length(lamda) r1(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===[pi/6,pi/2]================================ theta1=(-(pi/6)+e:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda)

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r2(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===============[pi/2,2pi/3]======================= ========== theta2=(-(pi/6)-e:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r3(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===============[5pi/6,7pi/6]====================== ============= theta3=((-pi/6)-e:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r4(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %=================[7pi/6,3pi/2]==================== theta4=((-pi/6)-e:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r5(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %===============[3pi/2,-pi/6]====================== =============== theta5=(-pi/6:0.01:pi/6); for j=1:length(I1) for i=1:length(lamda) r6(i,j)=(sigmac./6).*(-lamda(i)+sqrt(lamda(i).^2+12.*(mb*I1(j)/(3*sigmac)) +S)); end end %-------------------------------------------------- ------------------------ figure(1) % 2-D Generalised Priest trace on the deviatoric pl ane. for h=1:length(I1) [X,Y] = pol2cart(theta,r1(:,h)'); plot(X,Y) hold on [X1,Y1] = pol2cart(theta1+pi/3,r2(:,h)'); plot(X1,Y1) hold on [X2,Y2] = pol2cart(theta2+2*pi/3,r3(:,h)'); plot(X2,Y2) hold on [X3,Y3] = pol2cart(theta3+pi,r4(:,h)'); plot(X3,Y3) hold on [X4,Y4] = pol2cart(theta4+4*pi/3,r5(:,h)'); plot(X4,Y4) hold on [X5,Y5] = pol2cart(theta5+5*pi/3,r6(:,h)'); plot(X5,Y5) end

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xlabel( '\sigma_2d (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1d (Mpa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); R1=[1/sqrt(2) 0 -1/sqrt(2);-1/sqrt(6) 2/sqrt(6) -1/ sqrt(6);1/sqrt(3) 1/sqrt(3) 1/sqrt(3)]; R=inv(R1); figure(2) % 3-D Generalised Priest surface in stress space. for h=1:length(I1) [X,Y] = pol2cart(theta,sqrt(2)*r1(:,h)'); I16=(((((6*r1(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z1=I16; A1=R*[X;Y;z1/sqrt(3)]; P1=plot3(A1(1,:),A1(2,:),A1(3,:), 'r' ) %set(P1,'color',[1,0.7344,0]); hold on [X1,Y1] = pol2cart(theta1+pi/3,sqrt(2)*r2(:,h)' ); I16=(((((6*r2(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z2=I16; B=R*[X1;Y1;z2/sqrt(3)]; P2=plot3(B(1,:),B(2,:),B(3,:), 'r' ) %set(P2,'color',[1,0.7344,0]); hold on [X2,Y2] = pol2cart(theta2+2*pi/3,sqrt(2)*r3(:,h )'); I16=(((((6*r3(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z3=I16; C=R*[X2;Y2;z3/sqrt(3)]; P3=plot3(C(1,:),C(2,:),C(3,:), 'r' ) %set(P3,'color',[1,0.7344,0]); hold on [X3,Y3] = pol2cart(theta3+pi,sqrt(2)*r4(:,h)'); I16=(((((6*r4(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z4=I16; D=R*[X3;Y3;z4/sqrt(3)]; P4=plot3(D(1,:),D(2,:),D(3,:), 'r' ) %set(P4,'color',[1,0.7344,0]); hold on [X4,Y4] = pol2cart(theta4+4*pi/3,sqrt(2)*r5(:,h )'); I16=(((((6*r5(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z5=I16; E=R*[X4;Y4;z5/sqrt(3)]; P5=plot3(E(1,:),E(2,:),E(3,:), 'r' ) %set(P5,'color',[1,0.7344,0]); hold on [X5,Y5] = pol2cart(theta5+5*pi/3,sqrt(2)*r6(:,h )'); I16=(((((6*r6(:,h)'/sigmac)+lamda).^2-lamda.^2) ./(12))-S)*3*sigmac/mb; z6=I16; F=R*[X5;Y5;z6/sqrt(3)]; P6=plot3(F(1,:),F(2,:),F(3,:), 'r' ) %set(P6,'color',[1,0.7344,0]); end % Moving axes to the origin of coordinate system. plot3(get(gca, 'XLim' ),[0 0],[0 0], 'k' ); plot3([0 0],[0 0],get(gca, 'ZLim' ), 'k' ); plot3([0 0],get(gca, 'YLim' ),[0 0], 'k' );

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X=get(gca, 'Xtick' ); Y=get(gca, 'Ytick' ); Z=get(gca, 'Ztick' ); XL=get(gca, 'XtickLabel' ); YL=get(gca, 'YtickLabel' ); ZL=get(gca, 'ZtickLabel' ); % REMOVE TICKS set(gca, 'Xtick' ,[]); set(gca, 'Ytick' ,[]); set(gca, 'Ztick' ,[]); % GET OFFSETS Xoff=diff(get(gca, 'XLim' ))./30; Yoff=diff(get(gca, 'YLim' ))./30; Zoff=diff(get(gca, 'ZLim' ))./30; xlabel( '\sigma_2 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) ylabel( '\sigma_1 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14) zlabel( '\sigma_3 (MPa)' , 'fontname' , 'times new roman' , 'fontsize' ,14)

analysis of rock performance under three-dimensional

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